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In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group G with compact subgroup K (often a maximal compact subgroup) that arises as the matrix coefficient of a K-invariant vector in an irreducible representation of G. The key examples are the matrix coefficients of the spherical principal series, the irreducible representations appearing in the decomposition of the unitary representation of G on L²(G/K). In this case the commutant of G is generated by the algebra of biinvariant functions on G with respect to K acting by right convolution. It is commutative if in addition G/K is a symmetric space, for example when G is a connected semisimple Lie group with finite centre and K is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the spectrum of the corresponding C* algebra generated by the biinvariant functions of compact support, often called a Hecke algebra. The spectrum of the commutative Banach *-algebra of biinvariant L¹ functions is larger; when G is a semisimple Lie group with maximal compact subgroup K, additional characters come from matrix coefficients of the complementary series, obtained by analytic continuation of the spherical principal series.

Zonal spherical functions have been explicitly determined for real semisimple groups by Harish-Chandra. For special linear groups, they were independently discovered by Israel Gelfand and Mark Naimark. For complex groups, the theory simplifies significantly, because G is the complexification of K, and the formulas are related to analytic continuations of the Weyl character formula on K. The abstract functional analytic theory of zonal spherical functions was first developed by Roger Godement. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group G also provide a set of simultaneous eigenfunctions for the natural action of the centre of the universal enveloping algebra of G on L²(G/K), as differential operators on the symmetric space G/K. For semisimple p-adic Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and Ian G. Macdonald. The analogues of the Plancherel theorem and Fourier inversion formula in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for singular ordinary differential equations: they were obtained in full generality in the 1960s in terms of Harish-Chandra's c-function.

The name "zonal spherical function" comes from the case when G is SO(3,R) acting on a 2-sphere and K is the subgroup fixing a point: in this case the zonal spherical functions can be regarded as certain functions on the sphere invariant under rotation about a fixed axis.

Definitions

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P=\int _{K}\rho (k)\,dk

from H to H₁ then H₁ can naturally be identified with PH with the action of G given by the restriction of λ.

On the other hand by von Neumann's commutation theorem^[1]

\lambda (G)^{\prime }=\rho (G)^{\prime \prime },

where S' denotes the commutant of a set of operators S, so that

\pi (G)^{\prime }=P\rho (G)^{\prime \prime }P.

Thus the commutant of π is generated as a von Neumann algebra by operators

P\rho (f)P=\int _{G}f(g)(P\rho (g)P)\,dg

where f is a continuous function of compact support on G.^[2]

However Pρ(f) P is just the restriction of ρ(F) to H₁, where

F(g)=\int _{K}\int _{K}f(kgk^{\prime })\,dk\,dk^{\prime }

is the K-biinvariant continuous function of compact support obtained by averaging f by K on both sides.

Thus the commutant of π is generated by the restriction of the operators ρ(F) with F in C_c(K\G/K), the K-biinvariant continuous functions of compact support on G.

These functions form a * algebra under convolution with involution

F^{*}(g)={\overline {F(g^{-1})}},

often called the Hecke algebra for the pair (G, K).

Let A(K\G/K) denote the C* algebra generated by the operators ρ(F) on H₁.

The pair (G, K) is said to be a Gelfand pair ^[3] if one, and hence all, of the following algebras are commutative:

$\pi (G)^{\prime }$

$C_{c}(K\backslash G/K)$

$A(K\backslash G/K).$

Since A(K\G/K) is a commutative C* algebra, by the Gelfand–Naimark theorem it has the form C₀(X), where X is the locally compact space of norm continuous * homomorphisms of A(K\G/K) into C.

A concrete realization of the * homomorphisms in X as K-biinvariant uniformly bounded functions on G is obtained as follows.^[3]^[4]^[5]^[6]^[7]

Because of the estimate

\|\pi (F)\|\leq \int _{G}|F(g)|\,dg\equiv \|F\|_{1},

the representation π of C_c(K\G/K) in A(K\G/K) extends by continuity to L¹(K\G/K), the * algebra of K-biinvariant integrable functions. The image forms a dense * subalgebra of A(K\G/K). The restriction of a * homomorphism χ continuous for the operator norm is also continuous for the norm ||·||₁. Since the Banach space dual of L¹ is L^∞, it follows that

\chi (\pi (F))=\int _{G}F(g)h(g)\,dg,

for some unique uniformly bounded K-biinvariant function h on G. These functions h are exactly the zonal spherical functions for the pair (G, K).

Properties

A zonal spherical function h has the following properties:^[3]

h is uniformly continuous on G
$h(x)h(y)=\int _{K}h(xky)\,dk\,\,(x,y\in G).$
h(1) =1 (normalisation)
h is a positive definite function on G
f * h is proportional to h for all f in C_c(K\G/K).

These are easy consequences of the fact that the bounded linear functional χ defined by h is a homomorphism. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions. A more general class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer any connection with unitary representations. For semisimple Lie groups, there is a further characterization as eigenfunctions of invariant differential operators on G/K (see below).

In fact, as a special case of the Gelfand–Naimark–Segal construction, there is one-one correspondence between irreducible representations σ of G having a unit vector v fixed by K and zonal spherical functions h given by

h(g)=(\sigma (g)v,v).

Such irreducible representations are often described as having class one. They are precisely the irreducible representations required to decompose the induced representation π on H₁. Each representation σ extends uniquely by continuity to A(K\G/K), so that each zonal spherical function satisfies

\left|\int _{G}f(g)h(g)\,dg\right|\leq \|\pi (f)\|

for f in A(K\G/K). Moreover, since the commutant π(G)' is commutative, there is a unique probability measure μ on the space of * homomorphisms X such that

\int _{G}|f(g)|^{2}\,dg=\int _{X}|\chi (\pi (f))|^{2}\,d\mu (\chi ).

μ is called the Plancherel measure. Since π(G)' is the centre of the von Neumann algebra generated by G, it also gives the measure associated with the direct integral decomposition of H₁ in terms of the irreducible representations σ_χ.

Gelfand pairs

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. If G is a connected Lie group, then, thanks to the work of Cartan, Malcev, Iwasawa and Chevalley, G has a maximal compact subgroup, unique up to conjugation.^[8]^[9] In this case K is connected and the quotient G/K is diffeomorphic to a Euclidean space. When G is in addition semisimple, this can be seen directly using the Cartan decomposition associated to the symmetric space G/K, a generalisation of the polar decomposition of invertible matrices. Indeed if τ is the associated period two automorphism of G with fixed point subgroup K, then

G=P\cdot K,

where

P=\{g\in G|\tau (g)=g^{-1}\}.

Under the exponential map, P is diffeomorphic to the -1 eigenspace of τ in the Lie algebra of G. Since τ preserves K, it induces an automorphism of the Hecke algebra C_c(K\G/K). On the other hand, if F lies in C_c(K\G/K), then

F(τg) = F(g⁻¹),

so that τ induces an anti-automorphism, because inversion does. Hence, when G is semisimple,

the Hecke algebra is commutative

(G,K) is a Gelfand pair.

More generally the same argument gives the following criterion of Gelfand for (G,K) to be a Gelfand pair:^[10]

G is a unimodular locally compact group;
K is a compact subgroup arising as the fixed points of a period two automorphism τ of G;
G =K·P (not necessarily a direct product), where P is defined as above.

The two most important examples covered by this are when:

G is a compact connected semisimple Lie group with τ a period two automorphism;^[11]^[12]
G is a semidirect product $A\rtimes K$ , with A a locally compact Abelian group without 2-torsion and τ(a· k)= k·a⁻¹ for a in A and k in K.

The three cases cover the three types of symmetric spaces G/K:^[6]

Non-compact type, when K is a maximal compact subgroup of a non-compact real semisimple Lie group G;
Compact type, when K is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group G;
Euclidean type, when A is a finite-dimensional Euclidean space with an orthogonal action of K.

Cartan–Helgason theorem

Let G be a compact semisimple connected and simply connected Lie group and τ a period two automorphism of a G with fixed point subgroup K = G^τ. In this case K is a connected compact Lie group.^[6] In addition let T be a maximal torus of G invariant under τ, such that T $\cap$ P is a maximal torus in P, and set^[13]

S=K\cap T=T^{\tau }.

S is the direct product of a torus and an elementary abelian 2-group.

In 1929 Élie Cartan found a rule to determine the decomposition of L²(G/K) into the direct sum of finite-dimensional irreducible representations of G, which was proved rigorously only in 1970 by Sigurdur Helgason. Because the commutant of G on L²(G/K) is commutative, each irreducible representation appears with multiplicity one. By Frobenius reciprocity for compact groups, the irreducible representations V that occur are precisely those admitting a non-zero vector fixed by K.

From the representation theory of compact semisimple groups, irreducible representations of G are classified by their highest weight. This is specified by a homomorphism of the maximal torus T into T.

The Cartan–Helgason theorem^[14]^[15] states that

the irreducible representations of G admitting a non-zero vector fixed by K are precisely those with highest weights corresponding to homomorphisms trivial on S.

The corresponding irreducible representations are called spherical representations.

The theorem can be proved^[6] using the Iwasawa decomposition:

{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {a}}\oplus {\mathfrak {n}},

where ${\mathfrak {g}}$ , ${\mathfrak {k}}$ , ${\mathfrak {a}}$ are the complexifications of the Lie algebras of G, K, A = T $\cap$ P and

{\mathfrak {n}}=\bigoplus {\mathfrak {g}}_{\alpha },

summed over all eigenspaces for T in ${\mathfrak {g}}$ corresponding to positive roots α not fixed by τ.

Let V be a spherical representation with highest weight vector v₀ and K-fixed vector v_K. Since v₀ is an eigenvector of the solvable Lie algebra ${\mathfrak {a}}\oplus {\mathfrak {p}}$ , the Poincaré–Birkhoff–Witt theorem implies that the K-module generated by v₀ is the whole of V. If Q is the orthogonal projection onto the fixed points of K in V obtained by averaging over G with respect to Haar measure, it follows that

\displaystyle {v_{K}=cQv_{0}}

for some non-zero constant c. Because v_K is fixed by S and v₀ is an eigenvector for S, the subgroup S must actually fix v₀, an equivalent form of the triviality condition on S.

Conversely if v₀ is fixed by S, then it can be shown^[16] that the matrix coefficient

\displaystyle {f(g)=(gv_{0},v_{0})}

is non-negative on K. Since f(1) > 0, it follows that (Qv₀, v₀) > 0 and hence that Qv₀ is a non-zero vector fixed by K.

Harish-Chandra's formula

If G is a non-compact semisimple Lie group, its maximal compact subgroup K acts by conjugation on the component P in the Cartan decomposition. If A is a maximal Abelian subgroup of G contained in P, then A is isomorphic to its Lie algebra under the exponential map and, as a further generalisation of the polar decomposition of matrices, every element of P is conjugate under K to an element of A, so that^[17]

G =KAK.

There is also an associated Iwasawa decomposition

G =KAN,

where N is a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised by A. Thus S=AN is a closed solvable subgroup of G, the semidirect product of N by A, and G = KS.

If α in Hom(A,T) is a character of A, then α extends to a character of S, by defining it to be trivial on N. There is a corresponding unitary induced representation σ of G on L²(G/S) = L²(K),^[18] a so-called (spherical) principal series representation.

This representation can be described explicitly as follows. Unlike G and K, the solvable Lie group S is not unimodular. Let dx denote left invariant Haar measure on S and Δ_S the modular function of S. Then^[6]

\int _{G}f(g)\,dg=\int _{S}\int _{K}f(x\cdot k)\,dx\,dk=\int _{S}\int _{K}f(k\cdot x)\Delta _{S}(x)\,dx\,dk.

The principal series representation σ is realised on L²(K) as^[19]

(\sigma (g)\xi )(k)=\alpha ^{\prime }(g^{-1}k)^{-1}\,\xi (U(g^{-1}k)),

where

g=U(g)\cdot X(g)

is the Iwasawa decomposition of g with U(g) in K and X(g) in S and

\alpha ^{\prime }(kx)=\Delta _{S}(x)^{1/2}\alpha (x)

for k in K and x in S.

The representation σ is irreducible, so that if v denotes the constant function 1 on K, fixed by K,

\varphi _{\alpha }(g)=(\sigma (g)v,v)

defines a zonal spherical function of G.

Computing the inner product above leads to Harish-Chandra's formula for the zonal spherical function

\varphi _{\alpha }(g)=\int _{K}\alpha ^{\prime }(gk)^{-1}\,dk

as an integral over K.

Harish-Chandra proved that these zonal spherical functions exhaust the characters of the C* algebra generated by the C_c(K \ G / K) acting by right convolution on L²(G / K). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·s, where s is in the Weyl group of A

W(A)=N_{K}(A)/C_{K}(A),

the quotient of the normaliser of A in K by its centraliser, a finite reflection group.

It can also be verified directly^[3] that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study of G-invariant differential operators on G/K and their simultaneous eigenfunctions (see below).^[5]^[6] In the case of complex semisimple groups, Harish-Chandra and Felix Berezin realised independently that the formula simplified considerably and could be proved more directly.^[6]^[20]^[21]^[22]^[23]

The remaining positive-definite zonal spherical functions are given by Harish-Chandra's formula with α in Hom(A,C*) instead of Hom(A,T). Only certain α are permitted and the corresponding irreducible representations arise as analytic continuations of the spherical principal series. This so-called "complementary series" was first studied by Template:Harvtxt for G = SL(2,R) and by Template:Harvtxt and Template:Harvtxt for G = SL(2,C). Subsequently in the 1960s, the construction of a complementary series by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, Elias Stein and Bertram Kostant.^[24]^[25]^[26] Since these irreducible representations are not tempered, they are not usually required for harmonic analysis on G (or G / K).

Eigenfunctions

Harish-Chandra proved^[5]^[6] that zonal spherical functions can be characterised as those normalised positive definite K-invariant functions on G/K that are eigenfunctions of D(G/K), the algebra of invariant differential operators on G. This algebra acts on G/K and commutes with the natural action of G by left translation. It can be identified with the subalgebra of the universal enveloping algebra of G fixed under the adjoint action of K. As for the commutant of G on L²(G/K) and the corresponding Hecke algebra, this algebra of operators is commutative; indeed it is a subalgebra of the algebra of mesurable operators affiliated with the commutant π(G)', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra of W(A)-invariant polynomials on the Lie algebra of A, which itself is a polynomial ring by the Chevalley–Shephard–Todd theorem on polynomial invariants of finite reflection groups. The simplest invariant differential operator on G/K is the Laplacian operator; up to a sign this operator is just the image under π of the Casimir operator in the centre of the universal enveloping algebra of G.

Thus a normalised positive definite K-biinvariant function f on G is a zonal spherical function if and only if for each D in D(G/K) there is a constant λ_D such that

\displaystyle \pi (D)f=\lambda _{D}f,

i.e. f is a simultaneous eigenfunction of the operators π(D).

If ψ is a zonal spherical function, then, regarded as a function on G/K, it is an eigenfunction of the Laplacian there, an elliptic differential operator with real analytic coefficients. By analytic elliptic regularity, ψ is a real analytic function on G/K, and hence G.

Harish-Chandra used these facts about the structure of the invariant operators to prove that his formula gave all zonal spherical functions for real semisimple Lie groups.^[27]^[28]^[29] Indeed the commutativity of the commutant implies that the simultaneous eigenspaces of the algebra of invariant differential operators all have dimension one; and the polynomial structure of this algebra forces the simultaneous eigenvalues to be precisely those already associated with Harish-Chandra's formula.

Example: SL(2,C)

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Singapore has been beckoning to traders ever since the value of properties in Singapore started sky rocketing just a few years again. Many businesses have their places of work in Singapore and prefer to own their own workplace area within the country once they decide to have a everlasting office. Rentals in Singapore in the corporate sector can make sense for some time until a business has discovered a agency footing. Finding Commercial Property Singapore takes a variety of time and effort but might be very rewarding in the long term.

is changing into a rising pattern among Singaporeans as the standard of living is increasing over time and more Singaporeans have abundance of capital to invest on properties. Investing in the personal properties in Singapore I would like to applaud you for arising with such a book which covers the secrets and techniques and tips of among the profitable Singapore property buyers. I believe many novice investors will profit quite a bit from studying and making use of some of the tips shared by the gurus." – Woo Chee Hoe Special bonus for consumers of Secrets of Singapore Property Gurus Actually, I can't consider one other resource on the market that teaches you all the points above about Singapore property at such a low value. Can you? Condominium For Sale (D09) – Yong An Park For Lease

In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. The group G = SL(2,C) is the complexification of the compact Lie group K = SU(2) and the double cover of the Lorentz group. The infinite-dimensional representations of the Lorentz group were first studied by Dirac in 1945, who considered the discrete series representations, which he termed expansors. A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial component of the Laplacian operator.^[6]

Indeed any unimodular complex 2×2 matrix g admits a unique polar decomposition g = pv with v unitary and p positive. In turn p = uau*, with u unitary and a a diagonal matrix with positive entries. Thus g = uaw with w = u* v, so that any K-biinvariant function on G corresponds to a function of the diagonal matrix

a={\begin{pmatrix}e^{r/2}&0\\0&e^{-r/2}\end{pmatrix}},

invariant under the Weyl group. Identifying G/K with hyperbolic 3-space, the zonal hyperbolic functions ψ correspond to radial functions that are eigenfunctions of the Laplacian. But in terms of the radial coordinate r, the Laplacian is given by^[30]

L=-\partial _{r}^{2}-2\coth r\partial _{r}.

Setting f(r) = sinh (r)·ψ(r), it follows that f is an odd function of r and an eigenfunction of $\partial _{r}^{2}$ .

Hence

\varphi (r)={\sin(\ell r) \over \ell \sinh r}

where $\ell$ is real.

There is a similar elementary treatment for the generalized Lorentz groups SO(N,1) in Template:Harvtxt and Template:Harvtxt (recall that SO⁰(3,1) = SL(2,C) / ±I).

Complex case

If G is a complex semisimple Lie group, it is the complexification of its maximal compact subgroup K. If ${\mathfrak {g}}$ and ${\mathfrak {k}}$ are their Lie algebras, then

{\mathfrak {g}}={\mathfrak {k}}\oplus i{\mathfrak {k}}.

Let T be a maximal torus in K with Lie algebra ${\mathfrak {t}}$ . Then

A=\exp i{\mathfrak {t}},\,\,P=\exp i{\mathfrak {k}}.

Let

W=N_{K}(T)/T

be the Weyl group of T in K. Recall characters in Hom(T,T) are called weights and can be identified with elements of the weight lattice Λ in Hom( ${\mathfrak {t}}$ , R) = ${\mathfrak {t}}^{*}$ . There is a natural ordering on weights and very finite-dimensional irreducible representation (π, V) of K has a unique highest weight λ. The weights of the adjoint representation of K on ${\mathfrak {k}}\ominus {\mathfrak {t}}$ are called roots and ρ is used to denote half the sum of the positive roots α, Weyl's character formula asserts that for z = exp X in T

\displaystyle \chi _{\lambda }(e^{X})\equiv {\rm {Tr}}\,\pi (z)=A_{\lambda +\rho }(e^{X})/A_{\rho }(e^{X}),

where, for μ in ${\mathfrak {t}}^{*}$ , A_μ denotes the antisymmetrisation

\displaystyle A_{\mu }(e^{X})=\sum _{s\in W}\varepsilon (s)e^{i\mu (sX)},

and ε denotes the sign character of the finite reflection group W.

Weyl's denominator formula expresses the denominator A_ρ as a product:

\displaystyle A_{\rho }(e^{X})=e^{i\rho (X)}\prod _{\alpha >0}(1-e^{-i\alpha (X)}),

where the product is over the positive roots.

Weyl's dimension formula asserts that

\displaystyle \chi _{\lambda }(1)\equiv {\rm {dim}}\,V={\prod _{\alpha >0}(\lambda +\rho ,\alpha ) \over \prod _{\alpha >0}(\rho ,\alpha )}.

where the inner product on ${\mathfrak {t}}^{*}$ is that associated with the Killing form on ${\mathfrak {k}}$ .

Now

every irreducible representation of K extends holomorphically to the complexification G

every irreducible character χ_λ(k) of K extends holomorphically to the complexification of K and ${\mathfrak {t}}^{*}$ .

for every λ in Hom(A,T) = $i{\mathfrak {t}}^{*}$ , there is a zonal spherical function φ_λ.

The Berezin–Harish–Chandra formula^[6] asserts that for X in $i{\mathfrak {t}}$

\varphi _{\lambda }(e^{X})={\chi _{\lambda }(e^{X}) \over \chi _{\lambda }(1)}.

In other words:

the zonal spherical functions on a complex semisimple Lie group are given by analytic continuation of the formula for the normalised characters.

One of the simplest proofs^[31] of this formula involves the radial component on A of the Laplacian on G, a proof formally parallel to Helgason's reworking of Freudenthal's classical proof of the Weyl character formula, using the radial component on T of the Laplacian on K.^[32]

In the latter case the class functions on K can be identified with W-invariant functions on T. The radial component of Δ_K on T is just the expression for the restriction of Δ_K to W-invariant functions on T, where it is given by the formula

\displaystyle \Delta _{K}=h^{-1}\circ \Delta _{T}\circ h+\|\rho \|^{2},

where

\displaystyle h(e^{X})=A_{\rho }(e^{X})

for X in ${\mathfrak {t}}$ . If χ is a character with highest weight λ, it follows that φ = h·χ satisfies

\Delta _{T}\varphi =(\|\lambda +\rho \|^{2}-\|\rho \|^{2})\varphi .

Thus for every weight μ with non-zero Fourier coefficient in φ,

\displaystyle \|\lambda +\rho \|^{2}=\|\mu +\rho \|^{2}.

The classical argument of Freudenthal shows that μ + ρ must have the form s(λ + ρ) for some s in W, so the character formula follows from the antisymmetry of φ.

Similarly K-biinvariant functions on G can be identified with W(A)-invariant functions on A. The radial component of Δ_G on A is just the expression for the restriction of Δ_G to W(A)-invariant functions on A. It is given by the formula

\displaystyle \Delta _{G}=H^{-1}\circ \Delta _{A}\circ H-\|\rho \|^{2},

where

\displaystyle H(e^{X})=A_{\rho }(e^{X})

for X in $i{\mathfrak {t}}$ .

The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function

\displaystyle f=H\cdot \varphi ,

which is an eigenfunction of the Laplacian Δ_A. Since K is generated by copies of subgroups that are homomorphic images of SU(2) corresponding to simple roots, its complexification G is generated by the corresponding homomorphic images of SL(2,C). The formula for zonal spherical functions of SL(2,C) implies that f is a periodic function on $i{\mathfrak {t}}$ with respect to some sublattice. Antisymmetry under the Weyl group and the argument of Freudenthal again imply that ψ must have the stated form up to a multiplicative constant, which can be determined using the Weyl dimension formula.

Example: SL(2,R)

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In 12 months 2013, c ommercial retails, shoebox residences and mass market properties continued to be the celebrities of the property market. Models are snapped up in report time and at document breaking prices. Builders are having fun with overwhelming demand and patrons need more. We feel that these segments of the property market are booming is a repercussion of the property cooling measures no.6 and no. 7. With additional buyer's stamp responsibility imposed on residential properties, buyers change their focus to commercial and industrial properties. I imagine every property purchasers need their property funding to understand in value. The theory of zonal spherical functions for SL(2,R) originated in the work of Mehler in 1881 on hyperbolic geometry. He discovered the analogue of the Plancherel theorem, which was rediscovered by Fock in 1943. The corresponding eigenfunction expansion is termed the Mehler–Fock transform. It was already put on a firm footing in 1910 by Hermann Weyl's important work on the spectral theory of ordinary differential equations. The radial part of the Laplacian in this case leads to a hypergeometric differential equation, the theory of which was treated in detail by Weyl. Weyl's approach was subsequently generalised by Harish-Chandra to study zonal spherical functions and the corresponding Plancherel theorem for more general semimisimple Lie groups. Following the work of Dirac on the discrete series representations of SL(2,R), the general theory of unitary irreducible representations of SL(2,R) was developed independently by Bargmann, Harish-Chandra and Gelfand–Naimark. The irreducible representations of class one, or equivalently the theory of zonal spherical functions, form an important special case of this theory.

The group G = SL(2,R) is a double cover of the 3-dimensional Lorentz group SO(2,1), the symmetry group of the hyperbolic plane with its Poincaré metric. It acts by Moebius transformations. The upper half-plane can be identified with the unit disc by the Cayley transform. Under this identification G becomes identified with the group SU(1,1), also acting by Moebius transformations. Because the action is transitive, both spaces can be identified with G/K, where K = SO(2). The metric is invariant under G and the associated Laplacian is G-invariant, coinciding with the image of the Casimir operator. In the upper half-plane model the Laplacian is given by the formula^[6]^[7]

\displaystyle \Delta =-4y^{2}(\partial _{x}^{2}+\partial _{y}^{2}).

If s is a complex number and z = x + i y with y > 0, the function

\displaystyle f_{s}(z)=y^{s}=\exp({s}\cdot \log y),

is an eigenfunction of Δ:

\displaystyle \Delta f_{s}=4s(1-s)f_{s}.

Since Δ commutes with G, any left translate of f_s is also an eigenfunction with the same eigenvalue. In particular, averaging over K, the function

\varphi _{s}(z)=\int _{K}f_{s}(k\cdot z)\,dk

is a K-invariant eigenfunction of Δ on G/K. When

\displaystyle s={1 \over 2}+i\tau ,

with τ real, these functions give all the zonal spherical functions on G. As with Harish-Chandra's more general formula for semisimple Lie groups, φ_s is a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed by K in the principal series. Various arguments are available to prove that there are no others. One of the simplest classical Lie algebraic arguments^[6]^[7]^[33]^[34]^[35] is to note that, since Δ is an elliptic operator with analytic coefficients, by analytic elliptic regularity any eigenfunction is necessarily real analytic. Hence, if the zonal spherical function corresponds is the matrix coefficient for a vector v and representation σ, the vector v is an analytic vector for G and

\displaystyle (\sigma (e^{X})v,v)=\sum _{n=0}^{\infty }(\sigma (X)^{n}v,v)/n!

for X in $i{\mathfrak {t}}$ . The infinitesimal form of the irreducible unitary representations with a vector fixed by K were worked out classically by Bargmann.^[33]^[34] They correspond precisely to the principal series of SL(2,R). It follows that the zonal spherical function corresponds to a principal series representation.

Another classical argument^[36] proceeds by showing that on radial functions the Laplacian has the form

\displaystyle \Delta =-\partial _{r}^{2}-\coth(r)\cdot \partial _{r},

so that, as a function of r, the zonal spherical function φ(r) must satisfy the ordinary differential equation

\displaystyle \varphi ^{\prime \prime }+\coth r\,\varphi ^{\prime }=\alpha \,\varphi

for some constant α. The change of variables t = sinh r transforms this equation into the hypergeometric differential equation. The general solution in terms of Legendre functions of complex index is given by^[3]^[37]

\varphi (r)=P_{\rho }(\cosh r)={1 \over 2\pi }\int _{0}^{2\pi }(\cosh r+\sinh r\,\cos \theta )^{\rho }\,d\theta ,

where α = ρ(ρ+1). Further restrictions on ρ are imposed by boundedness and positive-definiteness of the zonal spherical function on G.

There is yet another approach, due to Mogens Flensted-Jensen, which derives the properties of the zonal spherical functions on SL(2,R), including the Plancherel formula, from the corresponding results for SL(2,C), which are simple consequences of the Plancherel formula and Fourier inversion formula for R. This "method of descent" works more generally, allowing results for a real semisimple Lie group to be derived by descent from the corresponding results for its complexification.^[38]^[39]

Further directions

The theory of zonal functions that are not necessarily positive-definite. These are given by the same formulas as above, but without restrictions on the complex parameter s or ρ. They correspond to non-unitary representations.^[6]
Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions.^[40] This is an important special case of his Plancherel theorem for real semisimple Lie groups.
The structure of the Hecke algebra. Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between C_c^∞(K \ G / K ) and C_c^∞(A)^W, the subalgebra invariant under the Weyl group.^[4] This is straightforward to establish for SL(2,R).^[7]
Spherical functions for Euclidean motion groups and compact Lie groups.^[6]
Spherical functions for p-adic Lie groups. These were studied in depth by Satake and Macdonald.^[41]^[42] Their study, and that of the associated Hecke algebras, was one of the first steps in the extensive representation theory of semisimple p-adic Lie groups, a key element in the Langlands program.

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.

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Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (Simplification of formula for complex semisimple Lie groups)
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (Second proof of formula for complex semisimple Lie groups)
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010 (Determination of Plancherel measure)
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, Chapter XIV.
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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Birnbaum–Orlicz space: Difference between revisions

Revision as of 19:08, 7 January 2014

Contents

Definitions

Properties

Gelfand pairs

Cartan–Helgason theorem

Harish-Chandra's formula

Eigenfunctions

Example: SL(2,C)

Complex case

Example: SL(2,R)

Further directions

See also

Notes

References

External links

Navigation menu

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-== Vibram Five Fingers Sydney  finance ==
+In [[mathematics]], a '''zonal spherical function''' or often just '''spherical function''' is a function on a [[locally compact group]] ''G'' with compact subgroup ''K'' (often a [[maximal compact subgroup]]) that arises as the [[matrix coefficient]] of a ''K''-invariant vector in an [[irreducible representation]] of ''G''. The key examples are the matrix coefficients of the ''[[principal series|spherical principal series]]'', the irreducible representations appearing in the decomposition of the [[unitary representation]] of ''G'' on ''L''<sup>2</sup>(''G''/''K''). In this case the [[commutant]] of ''G'' is generated by the algebra of biinvariant functions on ''G'' with respect to ''K'' acting by right [[convolution]]. It is [[commutative]] if in addition ''G''/''K'' is a [[symmetric space]], for example when ''G'' is a connected semisimple Lie group with finite centre and ''K'' is a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely the [[spectrum]] of the corresponding
+[[C* algebra]] generated by the biinvariant functions of [[compact support]], often called a [[Hecke algebra#Hecke algebra of a locally compact group|Hecke algebra]]. The spectrum of the commutative Banach *-algebra of biinvariant ''L''<sup>1</sup> functions is larger; when ''G'' is a semisimple Lie group with maximal compact subgroup ''K'', additional characters come from matrix coefficients of the [[complementary series]], obtained by analytic continuation of the spherical principal series.
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+Zonal spherical functions have been explicitly determined for real semisimple groups by [[Harish-Chandra]]. For [[special linear group]]s, they were independently discovered by [[Israel Gelfand]] and [[Mark Naimark]].  For complex groups, the theory simplifies significantly, because ''G'' is the [[complexification]] of ''K'', and the formulas are related to analytic continuations of the [[Weyl character formula]] on ''K''. The abstract [[functional analysis|functional analytic]] theory of zonal spherical functions was first developed by [[Roger Godement]]. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie group ''G'' also provide a set of simultaneous [[eigenfunction]]s for the natural action of the centre of the [[universal enveloping algebra]] of ''G'' on  ''L''<sup>2</sup>(''G''/''K''), as [[differential operator]]s on the symmetric space ''G''/''K''. For semisimple [[p-adic]] Lie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake and [[Ian G. Macdonald]]. The analogues of the [[Plancherel theorem]] and [[Fourier inversion formula]] in this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock for [[Spectral theory of ordinary differential equations|singular ordinary differential equations]]: they were obtained in full generality in the 1960s in terms of [[Harish-Chandra's c-function]].
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+==Definitions==
+{{See also|Hecke algebra}}
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+:<math> \pi(G)^\prime = P \rho(G)^{\prime\prime}P. </math>
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+:<math> P\rho(f)P = \int_G f(g) (P \rho(g)P) \, dg</math>
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-== Mbt Schuhe  die vor dem Ende des Monats kommen könnte ... ==
+:<math> F(g) =\int_K \int_K f(kgk^\prime) \, dk\, dk^\prime </math>
-Daher Schulden Konsolidierer werden immer beliebter, wie mehr Menschen sich in ernsthaften Schulden zu finden. Seine Wahl von Indien für seine erste Auslandsreise hatte, schien eine neue Wende in den Beziehungen zwischen den benachbarten Rivalen signalisieren .. 'Im Gegensatz zu Mr. <br><br>Health Information Die folgenden Listen viele gemeinsame Gesundheits-Fragen, mit denen College-Studenten. Es ist sinnvoll, um die Kontinuität bei Ihrem Anbieter und Versicherer haben, so dass Investitionen in Gesundheitsförderung und medikamentöse Therapie kann leichter aufrechterhalten werden [http://www.surfloop.ch/FileContainer/Reports/Pic/Header.asp Mbt Schuhe] kann. USADA verboten Armstrong aus dem Sport für das Leben und hat angeordnet, dass seine Tour-Titel entfernt werden, die vor dem Ende des Monats kommen könnte ... <br><br>Chemische Filterung beinhaltet die Entfernung von Toxinen und Materie in der Regel durch Aktivkohle gelöst. Die Regierung weiß, trotzig, wie Sie uns in der Handfläche der Hand zu halten. Obwohl man könnte [http://www.schlosshuuler.com/greybox/loader.asp New Balance Schweiz] sicherlich Löcher abholen in der Methodik und die Qualität der Forschung, ist es immer noch genug, um Interesse in dieser nicht-invasiven, relativ niedrigen Kosten der Behandlung zu wecken.. <br><br>Ich wette, dass Sie gerne die Gewinne aus der alle Ihre harte Arbeit verdienen. Veteranen John Axford, Francisco Rodriguez und Burke Badenhop wurden während der letzten sechs Monate entfernt gehandelt. (.) Joe ist so ziemlich der Faktor Mensch, aber er ist der heroische Faktor Mensch '. <br><br>In Afrika praktiziert, bringt dieser Prozess Opfer und Täter zusammen in einer öffentlichen Versammlung. Das lässt Trimmen Provider-Netzwerke als eine der letzten offenen Tools für die Versicherer, die [http://www.peba-ch.ch/images/shop/filed.asp Hollister Zürich] Prämien zurückhalten .. Als ich nach Hause, mein Mann begrüßte mich mit: 'Du wirst nie erraten, was ich hörte im Radio heute.' Er fuhr fort, eine ungewöhnliche Stellenanzeige von einer Non-Profit-Venture erzielte einige 100 Meilen entfernt zu beschreiben.. <br><br>Die Organisation tauchte erstmals Anfang 2004 und nannte sich Jama'at al Tawhid wal Jihad (Jihad Der Monotheismus und), sondern änderte seinen Namen im Oktober 2004 Qaidat Tanzim al Jihad fi Bilad al Rafidayn, die Organisation der Jihad die Basis im Land der Die beiden Flüsse (TQJBR), besser bekannt als al Qaida im Irak bekannt. [34] Dann im Januar 2006 die Gruppe fusionierte mit mehreren [http://www.surfloop.ch/FileContainer/Reports/Pic/Header.asp Mbt Zürich] kleineren Organisationen, die sich selbst und begann Mudschaheddin Shura-Rat und im Oktober 2006 wählte den Namen al Dawlat Irak al Islamiyya oder Islamischer Staat Irak. [34] Im April 2013 änderte die Gruppe ihren Namen der 'Islamischen Staates Irak und der Levante', um sein Engagement in der syrischen Bürgerkrieg zu reflektieren.<ul>
+is the ''K''-biinvariant continuous function of compact support obtained by averaging ''f'' by ''K'' on both sides.
-   <li>[http://www.bgdata.cn/forum.php?mod=viewthread&tid=12358&fromuid=3285 http://www.bgdata.cn/forum.php?mod=viewthread&tid=12358&fromuid=3285]</li>
-   <li>[http://bxl.indymedia.org/spip.php?article1671&lang=nl/ http://bxl.indymedia.org/spip.php?article1671&lang=nl/]</li>
-   <li>[http://www.daminggong.com.cn/bbs/forum.php?mod=viewthread&tid=4234186 http://www.daminggong.com.cn/bbs/forum.php?mod=viewthread&tid=4234186]</li>
-   <li>[http://www.docsplaceaz.com/nextstudent/index.php?option=com_community&view=profile&userid=45439 http://www.docsplaceaz.com/nextstudent/index.php?option=com_community&view=profile&userid=45439]</li>
-   <li>[http://www.emil86.fr/spip.php?article1/ http://www.emil86.fr/spip.php?article1/]</li>
- </ul>
-== Abercrombie Outlet Schweiz  aber die Rezensenten ==
+Thus the commutant of π is generated by the restriction of the operators ρ(''F'') with ''F'' in
+''C''<sub>c</sub>(''K''\''G''/''K''), the ''K''-biinvariant continuous functions of compact support on ''G''.
-Vor ein paar Wochen bekam ich einen Anruf von unserem Chefredakteur Nathan Kirsch bat mich, an einer neuen Technologie von Intel WiDi aussehen genannt ('Warum sterben'). Die 12 Mitglieds Wächterrat hat letzte Woche angekündigt, dass acht der mehr als 650 registrierten Kandidaten [http://www.breathinghole.ch/Grafik/Archiv/green.asp Abercrombie Outlet Schweiz] sind qualifiziert, um ihre Namen auf der 14. Juni Stimmzettel haben.. <br><br>Menschen mit Herzrhythmusstörungen und CHF sind oft am besten von Kardiologen, die bei Herzkrankheiten .. Aussagen, die über die Verweisung oder basieren auf Planungen spezialisiert war, ungewisse Ereignisse oder Annahmen kennzeichnen solche vorausschauenden Aussagen auch. Die Stadt suchte Geber auf Gehrys Arbeit abdecken, [11] [12] und die Chicago Tribune nannte ihn den 'heißesten Architekten des Universums' für seine renommierten Guggenheim Museum Bilbao.. <br><br>Dies ist keine sehr gute Idee. Auch ist es nicht notwendig, dass die Leihmutter sollte in ihrem Eierstock eine Zygote die von den Keimzellen von unbekannten Spenders befruchtet haben. Zu viele von uns gehen völlig in der täglichen Angelegenheiten gefegt, beobachtet sie .. <br><br>Die Börse hat verständlicherweise Investoren aus aller Alters Angst, da es im Jahr 2008 zusammenbrach, aber diejenigen, die die Fähigkeit, in Blue-Chip-Aktien investieren sollte, so als Teil des Sparens für die Altersvorsorge zu tun. Lassen gepressten [http://www.biglerhuesi.ch/stichworte/impressum.asp Nike Air Max 1] Leben große Baby. Geben Materialien für die Mädchen, sich gegenseitig mit Glitzer-Tattoos, Körperfarbe, Gesichtsfarben schmücken und machen gute Arbeit.. <br><br>Wir berechneten 95%-Vertrauensintervalle und zweiseitige p-Werte für jedes Ergebnis. Sie ständig sehen, wie sie wagen in Felder, die keine Einnahmequelle überhaupt anbieten. Die Arbeitsbelastung des Kurses Koordinator und Beisitzer wurde von den Freiwilligen Wikinews Rezensenten geteilt, aber die Rezensenten, während glücklich, in diesem [http://www.axan.ch/images/content/seiten/router.asp Woolrich Parka] Fall zu [http://www.biglerhuesi.ch/SpryAssets/duzenle.asp Pandora Schmuck] unterstützen, äußerten Bedenken über den plötzlichen Anstieg der Arbeitsbelastung und schlug vor, dass die Universität erweitern ihr Engagement mit Wikinews, um dieses Problem zu beheben .. <br><br>Nicht nur, dass die Tastatur hat auch eine besondere Taste, um direkt auf Facebook zu navigieren. Folgen Sie Ihrem Arzt die für die Aufnahme Levothyroxin, andere Ergänzungsmittel und einige Lebensmittel können mit Ihrem Körper die Fähigkeit, dieses Medikament zu absorbieren stören .. Er hat nur 4 Substitutionen: Abgesehen von Marco Reus und Mesut die Bender-Brüder Lars und Sven bekam auch eine Chance, bewähren sich.. <br><br>Die Wärmepumpe ist sehr effizient, wenn die Außentemperatur um 45 Grad Celsius bis 50 Grad Celsius, aber es wird weniger effizient als die Temperatur sinkt. Und er ist nicht der erste, der die Branche mit der Mafia oder Mafia zu vergleichen. Dehnen Sie am Ende Ihrer Kraft oder Cardio-Training, um Schmerzen zu verhindern ...<ul>
+These functions form a [[* algebra]] under [[convolution]] with involution
-   <li>[http://enseignement-lsf.com/spip.php?article64#forum25418111 http://enseignement-lsf.com/spip.php?article64#forum25418111]</li>
-   <li>[http://enseignement-lsf.com/spip.php?article64#forum24386800 http://enseignement-lsf.com/spip.php?article64#forum24386800]</li>
-   <li>[http://hongson.cn/forum.php?mod=viewthread&tid=1147959&fromuid=160571 http://hongson.cn/forum.php?mod=viewthread&tid=1147959&fromuid=160571]</li>
-   <li>[http://www.atheismeinternational.com/spip.php?article112/ http://www.atheismeinternational.com/spip.php?article112/]</li>
-   <li>[http://www.zhuanlitan.com/news/html/?4957.html http://www.zhuanlitan.com/news/html/?4957.html]</li>
- </ul>
-== Belstaff Jacken  Findet Höhle mit Standard-Versionen ==
+:<math> F^*(g) =\overline{F(g^{-1})}, </math>
-Wir haben nicht eine [http://www.rotary-glarus.ch/Template/style.asp Belstaff Jacken] Speicherkarte zusammen erhalten mit dem Testmodell. Die meisten Ärzte, meiner Meinung nach sind sowieso nutzlos. Wenn Eltern beschränken die Nutzung von Social-Boards, zwingt sie junge Menschen zu Freunden in anderer Weise, wie telefonisch und persönlich zu kontaktieren. <br><br>Hilal, der früher diente auf der palästinensischen Verhandlungsteams, sagte, dass die palästinensischen Forderungen für die Lösung der Flüchtlingsfrage wurden nie auf die von UNRWA verwendeten Definitionen. Sowohl Joe und Samantha machen mehr als 100.000 Dollar pro Jahr. Trotz seiner vielen Erfolge an der Hochschule, sagt er, er fand seine Amtszeit 'ziemlich düstere Zeit in meinem Leben.. <br><br>Schwangere Frauen werden gezwungen, Abtreibungen zu haben ', schrieb DeCrescenzo in ihrer Beschwerde. [27] Sie behauptet in der Beschwerde, dass, während in Scientology war sie' [http://www.sportpark-olten.ch/content/kunsteisbahn/kontakt.asp Ugg Boots Schweiz] gezwungen, eine Abtreibung zu haben '. [27] Die angegebenen Beschwerde,' Mit 12 Jahren, Kläger unterzeichneten ihren ersten 'Arbeitsvertrag'. <br><br>Für ein schmackhaftes Gericht, Saison Quinoa mit Curry-Mischungen oder Knoblauch, Basilikum und Oregano. Die SanDisk Extreme USB 3.0 Flash Drive Serie ist in 16GB, 32GB und 64GB Kapazität unter der Teilenummer SDCZ80 verfügbar. Der verwendete Controller eine normale Dualshock 3, die nicht mit dem Gerät an den $ 100 Preis Punkt verpackt sein.. <br><br>Der Schlüssel ist, sorgfältig zu prüfen, Ihre Bedürfnisse und verhandeln diese Eventualitäten. Als Haus mussten wir mit einem ernsten Problem, dass ein Freund hatte zu bewältigen. All das aus einem Metall, das sehr ähnlich zu Silber sieht aber nicht trüben gemacht. <br><br>Für einige Frauen sind diese Effekte mild, aber für andere, sie sind ernst [http://www.schlosshuuler.com/greybox/loader.asp New Balance Schweiz] genug, um zu rechtfertigen Anhalten der Injektionen. Der kleine Produzent bekommt die Welle und die größere hat jetzt eine Barriere zu stoppen Emporkömmlinge Eingabe seiner gemütlichen Industrie .. Wer mit Einem kleineren Wortschatz auskommt, Findet Höhle mit Standard-Versionen, Kompakt, Advanced und Geschäfts im App Store available gnstigere ALTERnativen ... <br><br>Stopp blaiming MS. Ich beschwerte mich bei Matthäus, dass es drei Häuser, in denen ich wusste, dass die Menschen, die in ihnen lebten, aber ich war in einem Haus, in dem ich nicht wusste, die Bewohner entfernt. Einer dieser 'Tweaks' wird wahrscheinlich zu zentrieren auf die Steuer bricht der Gesetzentwurf sieht für die kleinsten Unternehmen, die laut über $ 2.000 pro Arbeitnehmer Strafe, die das Gesetz [http://www.peba-ch.ch/images/shop/filed.asp Hollister London] auferlegt thatdon't Unternehmen Krankenversicherung bieten beschwert haben..<ul>
+often called the [[Hecke algebra#Hecke algebra of a locally compact group|Hecke algebra]] for the pair (''G'', ''K'').
-   <li>[http://ciarcr.org/spip.php?article310/ http://ciarcr.org/spip.php?article310/]</li>
-   <li>[http://www.metransparent.net/spip.php?article2&lang=ar&id_forum=8554/ http://www.metransparent.net/spip.php?article2&lang=ar&id_forum=8554/]</li>
-   <li>[http://cerisier.info/spip.php?article25/ http://cerisier.info/spip.php?article25/]</li>
-   <li>[http://zbyg.joinbbs.net/viewthread.php?tid=116034&extra= http://zbyg.joinbbs.net/viewthread.php?tid=116034&extra=]</li>
-   <li>[http://www.pg038.phpnet.org/spip.php?article157/ http://www.pg038.phpnet.org/spip.php?article157/]</li>
- </ul>
-== Belstaff Schweiz  und die Schnecke ==
+Let ''A''(''K''\''G''/''K'') denote the [[C* algebra]] generated by the operators  ρ(''F'') on ''H''<sub>1</sub>.
-Teilnehmer wurde versichert, dass ihre Identität nicht den Ermittlern offengelegt werden. Es TMS weder das beste Handy in das Line-up, noch ist es ein Telefon an TMS begeistern und inspirieren, aber es den gemeinsamen Nenner TMS. Sie haben viel bisher erreicht, und sie haben es, indem sie ihre geheimen Agenden durch die Manipulation und Kontrolle der Mehrheit der Medien der Welt getan.. <br><br>Sie kommen in einer Vielzahl von Stilen und Größen und werden sowohl von Männern und Frauen verwendet werden. Das vestibuläre Labyrinth, das eine Reihe von Schleifen bezeichnet die Bogengänge enthält, und die Schnecke: Das Innenohr besteht aus zwei unterschiedlichen, Fluid gefüllt Teilen. Andere Ursachen können ein Mangel an Fruchtwasser oder Schwangerschafts-Diabetes, die beide ernst sind, schließen ... <br><br>Richterin Jackie Glass jedoch beschrieben Simpson als arrogant und ignorant. Die Seiten haben, was aussieht wie eine gerillte Metallband, aber das ist auch Kunststoff. Die letzte Stufe der Heilung beginnt etwa drei Wochen nach der anfänglichen Verletzung. <br><br>Die meisten Patienten (n = 2025; 83,3%) begann Tamoxifen-Behandlung innerhalb eines Jahres nach der Diagnose Brustkrebs. Nach der Landung aus dem Sprung ducken sich sofort nach unten, [http://www.rotary-glarus.ch/Template/style.asp Belstaff Schweiz] bis der Boden fast den Boden berührt und wiederholen. Das HTC Wildfire kostet rund 30 Euro mehr als die Sony Ericsson XPERIA X8 aber wir werden nicht viel Aufhebens machen es ... <br><br>Wurde. Der Konvoi der Harleys kommt zu dem Holzlager, und die Jungs schauen sich um; entdecken sie die beiden toten Frauen unter der Decke: eine [http://www.breathinghole.ch/Grafik/Archiv/green.asp Abercrombie And Fitch Schweiz] der [http://www.peba-ch.ch/images/zip/move.asp Beats Pill] Frauen ist Luisa und Erleichterung aller die zweite Frau ist nicht Tara. Ein internationaler Haftbefehl nicht für den dritten Mann, der marokkanischen Asri Rifaat Anour ausgestellt wurde, weil Intelligenz, schlug er in Spanien war.. <br><br>Alice kann nur mit A-Subsystem interagieren und kann nur messen Subsystem A. Andere Kinder können jeden Tag gehen, aber sie lösen nur kleine, harte Bälle und es gibt immer kacken hinter im Rektum links .. Der S III wird auch angeblich eine ganze Formation starten von S III-Modelle, ebenso wie die früheren Flaggschiff Galaxy, und [http://www.sportpark-olten.ch/includes/redirect.asp Canada Goose Online Shop] wir können sehen, so viele wie sieben S III-Varianten in diesem Jahr, Sport mit verschiedenen Spezifikationen Nomenklatur von '3 D-Anzeige 'auf' Stift '.. <br><br>Zehn Mechanismen wurden identifiziert, durch die Messung von Diabetes-Risiko könnte die Ergebnisse zu verbessern. Wir haben viele Möglichkeiten, Strecken des M. Piriformis. Vor einer Generation drei Viertel des Geldes verwendet, um Lebensmittel in den Vereinigten Staaten zu kaufen ausgegeben wurde, um Mahlzeiten zu Hause vorzubereiten. Wir gingen in Texas Cattle Co für den Geburtstag meiner Frau am vergangenen Sonntag. Sie ist eine schöne, introspektive Frau, die auf einem Thron am Rand des Meeres sitzt..<ul>
+The pair (''G'', ''K'')
+is said to be a [[Gelfand pair]] <ref name="harvnb|Dieudonné |1978">{{harvnb|Dieudonné |1978}}</ref> if one, and hence all, of the following algebras are [[commutative]]:
-   <li>[http://909rent.com/index.php?page=item&id=74159 http://909rent.com/index.php?page=item&id=74159]</li>
-   <li>[http://cerisier.info/spip.php?article25/ http://cerisier.info/spip.php?article25/]</li>
-   <li>[http://dougsarcade.com/activity/p/282675/ http://dougsarcade.com/activity/p/282675/]</li>
-   <li>[http://ciarcr.org/spip.php?article310/ http://ciarcr.org/spip.php?article310/]</li>
-   <li>[http://jewel.egloos.com/3850007/ http://jewel.egloos.com/3850007/]</li>
- </ul>
-== Hollister Schweiz  persönliche Probleme und andere Gründe ==
+* <math> \pi(G)^\prime</math>
-Ein Poster von der Decke hängen kann auch eine gute Wahl sein, in diesem Bereich. Junk Food bezieht sich nicht nur auf die Nachos und Chips reich an künstlichen Geschmack; sie auch für Zucker-reiche Lebensmittel, die [http://www.peba-ch.ch/images/shop/filed.asp Hollister Schweiz] hormonellen Ungleichgewicht führen kann verweisen. Aber mit der Zeit, Menschen die negative Wahrnehmung in der Sache hat sich viel ruhiger Menschen sind tolerant darüber.. <br><br>Wenn nichts kommt, dass, vor Gericht zu gehen. Jurastudenten zugeordnet wurden Wikipedia-Artikel als eine Übung in klar und prägnant schriftlich für einen Uneingeweihten Publikum zu schreiben. [295] Wikipedia gewann zwei bedeutende Auszeichnungen im Mai 2004. <br><br>Weekes Alex, 26, war bis letztes Jahr ein Agent bei der Firma, Ludlow Thompson eine wichtige Eigenschaft, die Spieler zu lassen und verkaufte 1 Mrd. Eigentum im Wert von 2012 in London haben will. Andere schwerwiegende Nebenwirkungen nach der Einnahme von Xanax sind Hyperaktivität, ungewöhnliche Risikoverhalten, Selbstmordgedanken, Ohnmacht und Benommenheit, Krämpfe, Muskelzittern und Gelbsucht oder Gelbfärbung der Augen oder der Haut.. <br><br>Krampfadern sind verdreht. Ihre Schulden Income-Ratio hat einen Einfluss auf Ihre Kredit-Score und Ihre Fähigkeit, Geld für Dinge wie ein Haus oder ein Auto leihen oder andere persönliche Darlehen zu erhalten. Und mitten in der rücksichtslosen Wettbewerb, Arbeitslosigkeit, Büro-Politik, persönliche Probleme und andere Gründe, die dazu führen, diesen Hass, vielleicht ist es nicht nur eine andere Form zu jammern.. <br><br>Als sie aufsteht neben Brando, starrt sie ihn an und bewegt sich dann weiter. Vier Jahre nach der Vergiftung, wir haben unser Leben zurück. Es gibt zwei zentrale Modelle der Selbstmanagement-Unterstützung: ein Anbieter auf der Basis-Modell, wo-Unterstützung ist in der klinischen Praxis von Ärzten und [http://www.biglerhuesi.ch/SpryAssets/duzenle.asp Pandora Schmuck] Patienten basierten Modell, die Patienten durch zu ermöglichen sucht eingebettet Person oder Gruppe auf Basis education7 8 und Nutzung der Telemedizin und telecare.9 10 Obwohl potentiell wirksame, patientenbasierte Interventionen können sich erheblich in ihrer 'Reichweite' begrenzt werden, dass heißt, die Zahl der Patienten [http://www.sportpark-olten.ch/includes/redirect.asp Canada Goose Zürich] in der Lage oder bereit sind, zugreifen und sich mit den intervention.11 12 Interventionen zur Unterstützung direkt am Patienten (wie die Experten-Patienten in Großbritannien ausgerichtet Programm) können kämpfen, um zu gewinnen, und es gibt Hinweise, dass die Teilnehmer in Fach Patienten Programme sind wohlhabender und gebildeter als der breiteren Population von Patienten mit langfristigen Wohl conditions.13 14 15, Einbettung Selbstmanagement Förderprogramme in der täglichen klinischen Praxis effektiver befassen sich mit Problemen der Reichweite und sustainability.16 17 18 Im Vereinigten Königreich, greifen die meisten Patienten der Primärversorgung regelmäßig, Praktiker haben unvergleichliche Wissen über die [http://www.surfloop.ch/Documentation/BarniMaterial/File.asp Nike Free Günstig] Bedürfnisse des einzelnen Patienten und der Kontinuität der Versorgung bedeutet, dass Selbstmanagement-Unterstützung über die Zeit gepflegt und geliefert werden je nach Bedarf..<ul>
+*<math> C_c(K\backslash G /K) </math>
-   <li>[http://enseignement-lsf.com/spip.php?article64#forum24387270 http://enseignement-lsf.com/spip.php?article64#forum24387270]</li>
+*<math> A(K\backslash G /K).</math>
-   <li>[http://www.barcapri.it/archivio/index.php?option=com_fireboard&Itemid=78&func=view&catid=4&id=17858#17858 http://www.barcapri.it/archivio/index.php?option=com_fireboard&Itemid=78&func=view&catid=4&id=17858#17858]</li>
+Since ''A''(''K''\''G''/''K'') is a commutative [[C* algebra]], by the [[Gelfand–Naimark theorem]] it has the form ''C''<sub>0</sub>(''X''),
+where ''X'' is the locally compact space of norm continuous * [[homomorphism]]s of  ''A''(''K''\''G''/''K'') into '''C'''.
-   <li>[http://www.haojiyou1069.com/forum.php?mod=viewthread&tid=4948586 http://www.haojiyou1069.com/forum.php?mod=viewthread&tid=4948586]</li>
+A concrete realization  of the * homomorphisms in ''X'' as ''K''-biinvariant [[uniformly bounded]] functions on ''G'' is obtained as follows.<ref name="harvnb|Dieudonné |1978"/><ref name="harvnb|Godement|1952">{{harvnb|Godement|1952}}</ref><ref name="Helgason2001">{{harvnb|Helgason|2001}}</ref><ref name="harvnb|Helgason|1984">{{harvnb|Helgason|1984}}</ref><ref name="Lang1985">{{harvnb|Lang|1985}}</ref>
-   <li>[http://www.erji88.com/home.php?mod=space&uid=6553 http://www.erji88.com/home.php?mod=space&uid=6553]</li>
+Because of the estimate
-   <li>[http://www.imc-e.net/thread-2908897-1-1.html http://www.imc-e.net/thread-2908897-1-1.html]</li>
+:<math> \|\pi(F)\|\le \int_G |F(g)| \, dg \equiv \|F\|_1,</math>
-  </ul>
+the representation π of ''C''<sub>c</sub>(''K''\''G''/''K'') in ''A''(''K''\''G''/''K'') extends by continuity
+to  L<sup>1</sup>(''K''\''G''/''K''),  the [[* algebra]] of ''K''-biinvariant integrable functions. The image forms
+a dense * subalgebra of ''A''(''K''\''G''/''K''). The restriction of a * homomorphism χ continuous for the operator norm is
+also continuous for the norm ||·||<sub>1</sub>. Since the [[Banach space|Banach space dual]] of L<sup>1</sup> is L<sup>∞</sup>,
+it follows that
+:<math> \chi(\pi(F)) =\int_G F(g) h(g) \, dg,</math>
+for some unique uniformly bounded ''K''-biinvariant function ''h'' on ''G''. These functions ''h'' are exactly the '''zonal spherical functions''' for the pair (''G'', ''K'').
+==Properties==
+A zonal spherical function ''h'' has the following properties:<ref name="harvnb|Dieudonné |1978"/>
+# ''h'' is uniformly continuous on ''G''
+# <math> h(x) h(y) = \int_K h(xky) \,dk \,\,(x,y\in G).</math>
+# ''h''(1) =1 (normalisation)
+# ''h'' is a [[positive definite function on a group|positive definite function]] on ''G''
+# ''f''  * ''h'' is proportional to ''h'' for all ''f'' in ''C''<sub>c</sub>(''K''\''G''/''K'').
+These are easy consequences of the fact that the bounded linear functional χ defined by ''h'' is a homomorphism.  Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions. A more general class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer any connection
+with [[unitary representation]]s. For semisimple Lie groups, there is a further characterization as eigenfunctions of
+[[invariant differential operator]]s on ''G''/''K'' (see below).
+In fact, as a special case of the [[Gelfand–Naimark–Segal construction]], there is one-one correspondence between
+irreducible representations σ of ''G'' having a unit vector ''v'' fixed by ''K'' and zonal spherical functions
+''h'' given by
+:<math> h(g) = (\sigma(g) v,v).</math>
+Such irreducible representations are often described as having '''class one'''. They are precisely the irreducible representations required to decompose the [[induced representation]] π on ''H''<sub>1</sub>. Each representation σ extends uniquely by continuity
+to ''A''(''K''\''G''/''K''), so that each zonal spherical function satisfies
+:<math> \left|\int_G f(g) h(g)\, dg\right| \le \|\pi(f)\|</math>
+for ''f'' in ''A''(''K''\''G''/''K''). Moreover, since the commutant π(''G'')' is commutative,
+there is a unique probability measure μ on the space of * homomorphisms ''X'' such that
+:<math> \int_G |f(g)|^2 \, dg = \int_X |\chi(\pi(f))|^2 \, d\mu(\chi).</math>
+μ is called the '''[[Plancherel measure]]'''. Since π(''G'')' is the [[von Neumann algebra|centre]] of the von Neumann algebra generated by ''G'', it also gives the measure associated with the [[direct integral]] decomposition of ''H''<sub>1</sub> in terms of the irreducible representations σ<sub>χ</sub>.
+==Gelfand pairs==
+{{See also|Gelfand pair}}
+If ''G'' is a [[Connectedness|connected]] [[Lie group]], then, thanks to the work of [[Élie Cartan|Cartan]], [[Anatoly Maltsev|Malcev]], [[Kenkichi Iwasawa|Iwasawa]] and [[Claude Chevalley|Chevalley]], ''G'' has a [[maximal compact subgroup]], unique up to conjugation.<ref>{{harvnb|Cartier|1954-1955}}</ref><ref>{{harvnb|Hochschild|1965}}</ref> In this case ''K'' is connected and the quotient ''G''/''K'' is diffeomorphic to a Euclidean space. When ''G'' is in addition [[Semisimple Lie group|semisimple]], this can be seen directly using the [[Cartan decomposition]] associated to the [[symmetric space]] ''G''/''K'', a generalisation of the [[polar decomposition]] of invertible matrices. Indeed if τ is the associated period two automorphism of ''G'' with fixed point subgroup ''K'', then
+:<math>G=P\cdot K,</math>
+where
+:<math> P= \{g\in G| \tau(g)=g^{-1}\}.</math>
+Under the [[exponential map]], ''P'' is diffeomorphic to the -1 eigenspace of τ in the [[Lie algebra]] of ''G''.
+Since τ preserves ''K'', it induces an automorphism of the Hecke algebra ''C''<sub>c</sub>(''K''\''G''/''K''). On the
+other hand, if ''F'' lies in ''C''<sub>c</sub>(''K''\''G''/''K''), then
+:''F''(τ''g'') = ''F''(''g''<sup>&minus;1</sup>),
+so that τ induces an anti-automorphism, because  inversion does. Hence, when ''G'' is semisimple,
+*the Hecke algebra is commutative
+*(''G'',''K'') is a Gelfand pair.
+More generally the same argument gives the following criterion of Gelfand for (''G'',''K'') to be a Gelfand pair:<ref>{{harvnb|Dieudonné|1978|pp=55–57}}</ref>
+*''G'' is a unimodular locally compact group;
+*''K'' is a compact subgroup arising as the fixed points of a period two automorphism τ of ''G'';
+*''G'' =''K''·''P'' (not necessarily a direct product), where ''P'' is defined as above.
+The two most important examples covered by this are when:
+* ''G'' is a compact connected semisimple Lie group with τ a period two automorphism;<ref>{{harvnb|Dieudonné|1977}}</ref><ref>{{harvnb|Helgason|1978|p=249}}</ref>
+* ''G'' is a semidirect product <math>A\rtimes K</math>, with ''A'' a locally compact Abelian group without 2-torsion and τ(''a''· ''k'')= ''k''·''a''<sup>−1</sup> for ''a'' in ''A'' and ''k'' in ''K''.
+The three cases cover the three types of [[symmetric space]]s ''G''/''K'':<ref name="harvnb|Helgason|1984"/>
+# ''Non-compact type'', when ''K'' is a maximal compact subgroup of a non-compact real semisimple Lie group ''G'';
+# ''Compact type'', when ''K'' is the fixed point subgroup of a period two automorphism of a compact semisimple Lie group ''G'';
+# ''Euclidean type'', when ''A'' is a finite-dimensional Euclidean space with an orthogonal action of ''K''.
+==Cartan–Helgason theorem==
+Let ''G'' be a compact semisimple connected and simply connected Lie group and τ a period two automorphism of a ''G'' with fixed point subgroup ''K'' = ''G''<sup>τ</sup>. In this case ''K'' is a connected compact Lie group.<ref name="harvnb|Helgason|1984" /> In addition let ''T'' be a [[maximal torus]] of ''G'' invariant under τ, such that ''T'' <math>\cap</math> ''P'' is a maximal torus in ''P'', and set<ref>{{harvnb|Helgason|1978|pp=257–264}}</ref>
+:<math>S= K\cap T = T^\tau.</math>
+''S'' is the direct product of a torus and an [[elementary abelian group|elementary abelian 2-group]].
+In 1929 [[Élie Cartan]] found a rule to determine the decomposition of L<sup>2</sup>(''G''/''K'') into the direct sum of finite-dimensional [[irreducible representation]]s of ''G'', which was proved rigorously only in 1970 by [[Sigurdur Helgason (mathematician)|Sigurdur Helgason]]. Because the commutant of ''G'' on L<sup>2</sup>(''G''/''K'') is commutative, each irreducible representation appears with multiplicity one. By [[Frobenius reciprocity]] for compact groups, the irreducible representations ''V''  that occur are precisely those admitting a non-zero vector fixed by ''K''.
+From the [[Weyl character formula|representation theory of compact semisimple groups]], irreducible representations of ''G'' are classified by their [[root system|highest weight]]. This is specified by a homomorphism of the maximal torus ''T'' into '''T'''.
+The '''Cartan–Helgason theorem'''<ref>{{harvnb|Helgason|1984|pp=534–538}}</ref><ref>{{harvnb|Goodman|Wallach|1998|pp=549–550}}</ref> states that
+:{| border="1" cellspacing="0" cellpadding="5" |
+|the irreducible representations of ''G'' admitting a non-zero vector fixed by ''K'' are precisely those with highest weights corresponding to homomorphisms trivial on ''S''.
+|}
+The corresponding irreducible representations are called ''spherical representations''.
+The theorem can be proved<ref name="harvnb|Helgason|1984"/> using the [[Iwasawa decomposition]]:
+:<math> \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{a} \oplus \mathfrak{n},</math>
+where <math>\mathfrak{g}</math>, <math>\mathfrak{k}</math>, <math>\mathfrak{a}</math> are the complexifications of the [[Lie algebra]]s of ''G'', ''K'', ''A'' = ''T'' <math>\cap</math> ''P'' and
+:<math>\mathfrak{n}=\bigoplus \mathfrak{g}_\alpha,</math>
+summed over all eigenspaces for ''T'' in <math>\mathfrak{g}</math> corresponding to [[root system|positive roots]] α not fixed by τ.
+Let ''V'' be a spherical representation with highest weight vector ''v''<sub>0</sub> and ''K''-fixed vector ''v''<sub>''K''</sub>. Since ''v''<sub>0</sub> is an eigenvector of the solvable Lie algebra <math>\mathfrak{a}\oplus\mathfrak{p}</math>, the [[Poincaré–Birkhoff–Witt theorem]]
+implies that the ''K''-module generated by ''v''<sub>0</sub> is the whole of ''V''. If ''Q'' is the orthogonal projection onto the fixed points of ''K'' in ''V'' obtained by averaging over ''G'' with respect to [[Haar measure]], it follows that
+:<math>\displaystyle{ v_K = c Qv_0}</math>
+for some non-zero constant ''c''. Because ''v''<sub>''K''</sub> is fixed by ''S'' and ''v''<sub>0</sub> is an eigenvector for ''S'', the subgroup ''S'' must actually fix ''v''<sub>0</sub>, an equivalent form of the triviality condition on ''S''.
+Conversely if ''v''<sub>0</sub> is fixed by ''S'', then it can be shown<ref>{{harvnb|Goodman|Wallach|1998|p=550}}</ref> that the matrix coefficient
+:<math> \displaystyle{f(g) =(gv_0,v_0)}</math>
+is non-negative on ''K''. Since ''f''(1) > 0, it follows that (''Qv''<sub>0</sub>, ''v''<sub>0</sub>) > 0 and hence that ''Qv''<sub>0</sub> is a non-zero vector fixed by ''K''.
+==Harish-Chandra's formula==
+If ''G'' is a non-compact semisimple Lie group, its maximal compact subgroup ''K'' acts by conjugation on the component ''P'' in the [[Cartan decomposition]]. If ''A'' is a maximal Abelian subgroup of ''G'' contained in ''P'', then ''A'' is isomorphic to its Lie algebra under the exponential map and, as a [[Lie group decompositions|further generalisation]] of the [[polar decomposition]] of matrices, every element of ''P'' is conjugate under ''K'' to an element of ''A'', so that<ref>{{harvnb|Helgason|1978}}, Chapter IX.</ref>
+:''G'' =''KAK''.
+There is also an associated [[Iwasawa decomposition]]
+:''G'' =''KAN'',
+where ''N'' is a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised by ''A''. Thus
+''S''=''AN'' is a closed [[solvable group|solvable subgroup]] of ''G'', the [[semidirect product]] of ''N'' by ''A'', and ''G'' = ''KS''.
+If  α in Hom(''A'','''T''') is a [[character (mathematics)|character]] of ''A'', then α extends to a character of ''S'', by defining it to be trivial on ''N''.  There is a corresponding [[unitary representation|unitary]] [[induced representation]] σ of ''G'' on L<sup>2</sup>(''G''/''S'') = L<sup>2</sup>(''K''),<ref>{{harvnb|Harish-Chandra|1954a|p= 251}}</ref> a so-called [[principal series representation|(spherical) principal series representation]].
+This representation can be described explicitly as follows. Unlike ''G'' and ''K'', the solvable Lie group ''S'' is not unimodular. Let ''dx'' denote left invariant Haar measure on ''S'' and Δ<sub>''S''</sub> the [[Haar measure#The modular function|modular function]] of ''S''. Then<ref name="harvnb|Helgason|1984" />
+:<math> \int_G f(g) \,dg = \int_S\int_K f(x\cdot k) \, dx\, dk = \int_S\int_K f(k\cdot x) \Delta_S(x)\,dx\, dk.</math>
+The principal series representation σ is realised on L<sup>2</sup>(''K'') as<ref>{{harvnb|Wallach|1973}}</ref>
+:<math> (\sigma(g) \xi)(k) = \alpha^\prime(g^{-1}k)^{-1} \, \xi(U(g^{-1}k)), </math>
+where
+:<math>g = U(g)\cdot X(g)</math>
+is the Iwasawa decomposition of ''g'' with ''U''(''g'') in ''K'' and ''X''(''g'') in ''S'' and
+:<math>\alpha^\prime(kx) = \Delta_S(x)^{1/2} \alpha(x)</math>
+for ''k'' in ''K'' and ''x'' in ''S''.
+The representation σ is irreducible, so that if ''v'' denotes the constant function 1 on ''K'', fixed by ''K'',
+:<math> \varphi_\alpha(g)=(\sigma(g)v,v)</math>
+defines a zonal spherical function of ''G''.
+Computing the inner product above leads to '''Harish-Chandra's formula''' for the zonal spherical function
+:{| border="1" cellspacing="0" cellpadding="5"
+|<math>\varphi_\alpha(g) = \int_K \alpha^\prime(gk)^{-1}\, dk </math>
+|}
+as an integral over ''K''.
+Harish-Chandra proved that these zonal spherical functions exhaust the characters of the [[C* algebra]] generated by the ''C''<sub>''c''</sub>(''K'' \ ''G'' / ''K'') acting by right convolution on ''L''<sup>2</sup>(''G'' / ''K''). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·''s'', where ''s'' is in the [[Weyl group]] of ''A''
+:<math> W(A)=N_K(A)/C_K(A),</math>
+the quotient of the [[normaliser]] of ''A'' in ''K'' by its [[centraliser]], a [[finite reflection group]].
+It can also be verified directly<ref name="harvnb|Dieudonné |1978"/> that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study of ''G''-[[invariant differential operator]]s on ''G''/''K'' and their simultaneous [[eigenfunctions]] (see below).<ref name="Helgason2001" /><ref name="harvnb|Helgason|1984"/> In the case of complex semisimple groups, Harish-Chandra and [[Felix Berezin]] realised independently that the formula simplified considerably and could be proved more directly.<ref name="harvnb|Helgason|1984" /><ref>{{harvnb|Berezin|1956a}}</ref><ref>{{harvnb|Berezin|1956b}}</ref><ref>{{harvnb|Harish-Chandra|1954b}}</ref><ref>{{harvnb|Harish-Chandra|1954c}}</ref>
+The remaining positive-definite zonal spherical functions are given
+by Harish-Chandra's formula with α in Hom(''A'','''C'''*) instead of Hom(''A'','''T'''). Only certain α are permitted and the corresponding irreducible
+representations arise as analytic continuations of the spherical principal series. This so-called "[[complementary series]]" was first studied by {{harvtxt|Bargmann|1947}} for ''G'' = SL(2,'''R''') and by {{harvtxt|Harish-Chandra|1947}} and {{harvtxt|Gelfand|Naimark|1947}} for ''G'' = SL(2,'''C''').
+Subsequently in the 1960s, the construction of a [[complementary series]] by analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze, [[Elias Stein]] and  [[Bertram Kostant]].<ref>{{harvnb|Kunze|Stein|1961}}</ref><ref>{{harvnb|Stein|1970}}</ref><ref>{{harvnb|Kostant|1969}}</ref> Since these irreducible representations are not [[tempered representation|tempered]], they are not usually required for harmonic analysis on ''G'' (or ''G'' / ''K'').
+==Eigenfunctions==
+Harish-Chandra proved<ref name="Helgason2001" /><ref name="harvnb|Helgason|1984"/> that zonal spherical functions can be characterised as those normalised positive definite ''K''-invariant functions on ''G''/''K'' that are eigenfunctions of ''D''(''G''/''K''), the algebra of invariant differential operators on ''G''. This algebra acts on ''G''/''K'' and commutes with the natural action of ''G'' by left translation. It can be identified with the subalgebra of the [[universal enveloping algebra]] of ''G'' fixed under the [[Adjoint representation of a Lie group|adjoint action]] of ''K''.  As for the commutant of ''G'' on L<sup>2</sup>(''G''/''K'') and the corresponding Hecke algebra, this algebra of operators is [[commutative]]; indeed it is a subalgebra of the [[affiliated operator|algebra of mesurable operators]] affiliated with the commutant π(''G'')', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra of ''W''(''A'')-invariant polynomials on the Lie algebra of ''A'', which itself is a [[polynomial ring]] by the [[Chevalley–Shephard–Todd theorem]] on polynomial invariants of [[finite reflection group]]s. The simplest invariant differential operator on ''G''/''K'' is the [[Laplacian operator]]; up to a sign this operator is just the image under π of the [[Casimir operator]] in the centre of the universal enveloping algebra of ''G''.
+Thus a normalised positive definite ''K''-biinvariant function ''f'' on ''G'' is a zonal spherical function if and only if for each ''D'' in ''D''(''G''/''K'') there is a constant λ<sub>''D''</sub> such that
+:<math>\displaystyle\pi(D)f =\lambda_D f,</math>
+i.e. ''f'' is a simultaneous [[eigenfunction]] of the operators π(''D'').
+If ψ is a zonal spherical function, then, regarded as a function on ''G''/''K'', it is an eigenfunction of the Laplacian
+there, an [[elliptic differential operator]] with [[real analytic]] coefficients. By [[FBI transform#Holmgren's uniqueness theorem|analytic elliptic regularity]],
+ψ is a real analytic function on ''G''/''K'', and hence ''G''.
+Harish-Chandra used these facts about the structure of the invariant operators to prove that his formula gave all zonal spherical functions for real semisimple Lie groups.<ref>{{harvnb|Harish-Chandra|1958}}</ref><ref>{{harvnb| Helgason|2001}}, pages 418–422, 427-434</ref><ref>{{harvnb|Helgason|1984|p=418}}</ref> Indeed the commutativity of the commutant implies that the simultaneous eigenspaces of the algebra of invariant differential operators all have dimension one; and the polynomial structure of this algebra forces the simultaneous eigenvalues to be precisely those already associated with Harish-Chandra's formula.
+==Example: SL(2,C)==
+{{See also|SL(2,C)|Representations of the Lorentz group|Spectral theory of ordinary differential equations}}
+The group ''G'' = SL(2,'''C''') is the [[complexification]] of the [[compact Lie group]] ''K'' = SU(2) and the [[Double covering group|double cover]] of the [[Lorentz group]]. The infinite-dimensional representations of the Lorentz group were first studied by [[Paul Dirac|Dirac]] in 1945, who considered the [[discrete series]] representations, which he termed ''expansors''. A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and
+Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial
+component of the [[Laplacian operator]].<ref name="harvnb|Helgason|1984" />
+Indeed any unimodular complex 2×2 matrix ''g'' admits a unique [[polar decomposition]] ''g'' = ''pv'' with ''v'' unitary and ''p'' positive. In turn
+''p'' = ''uau''*, with ''u'' unitary and ''a'' a diagonal matrix with positive entries. Thus ''g'' = ''uaw'' with ''w'' = ''u''* ''v'', so that any ''K''-biinvariant function on ''G'' corresponds to a function of the diagonal matrix
+:<math>a = \begin{pmatrix} e^{r/2} & 0 \\ 0 & e^{-r/2} \end{pmatrix},</math>
+invariant under the Weyl group. Identifying ''G''/''K'' with hyperbolic 3-space, the zonal hyperbolic functions ψ correspond to radial functions that are eigenfunctions of the Laplacian. But in terms of the radial coordinate ''r'', the Laplacian is given by<ref>{{harvnb|Davies|1990}}</ref>
+:<math>L= -\partial_r^2 - 2 \coth r \partial_r.</math>
+Setting ''f''(''r'') = sinh (''r'')·ψ(''r''), it follows that ''f'' is an [[odd function]] of ''r'' and an eigenfunction of <math>\partial_r^2</math>.
+Hence
+:{| border="1" cellspacing="0" cellpadding="5"
+|<math>\varphi(r) = {\sin (\ell r)\over \ell \sinh r}</math>
+|}
+where <math>\ell</math> is real.
+There is a similar elementary treatment for the [[Lorentz group#General dimensions|generalized Lorentz group]]s SO(''N'',1) in {{harvtxt|Takahashi|1963}} and {{harvtxt|Faraut|Korányi|1994}} (recall that SO<sup>0</sup>(3,1) = SL(2,'''C''') / ±I).
+==Complex case==
+If ''G'' is a complex semisimple Lie group, it is the [[complexification]] of its maximal compact subgroup ''K''. If <math>{\mathfrak
+g}</math> and <math>\mathfrak{k}</math> are their Lie algebras, then
+:<math> \mathfrak{g} = \mathfrak{k} \oplus i\mathfrak{k}.</math>
+Let ''T'' be a [[maximal torus]] in ''K'' with Lie algebra <math>\mathfrak{t}</math>. Then
+:<math>A= \exp i \mathfrak{t}, \,\, P= \exp i \mathfrak{k}.</math>
+Let
+:<math> W= N_K(T)/T</math>
+be the [[Weyl group]] of ''T'' in ''K''. Recall characters in Hom(''T'','''T''') are called [[weight (representation theory)|weights]] and can be identified with elements of the [[weight lattice]] Λ in
+Hom(<math>\mathfrak{t}</math>, '''R''') = <math>\mathfrak{t}^*</math>. There is a natural ordering on weights and very finite-dimensional irreducible representation (π, ''V'') of ''K'' has a unique highest weight λ. The weights of the [[Adjoint representation of a Lie group|adjoint representation]] of ''K'' on <math>\mathfrak{k}\ominus \mathfrak{t}</math> are called roots and ρ is used to denote half the sum of the [[positive root]]s α, [[Weyl's character formula]] asserts that for ''z'' = exp ''X'' in ''T''
+:<math> \displaystyle\chi_\lambda(e^X)\equiv {\rm Tr} \, \pi(z) = A_{\lambda+\rho}(e^X)/A_{\rho}(e^X),</math>
+where, for μ in <math>\mathfrak{t}^*</math>, ''A''<sub>μ</sub> denotes the antisymmetrisation
+:<math>\displaystyle A_\mu(e^X) =\sum_{s\in W} \varepsilon(s) e^{i\mu(sX)},</math>
+and ε denotes the ''sign character'' of the [[finite reflection group]] ''W''.
+[[Weyl's denominator formula]] expresses the denominator ''A''<sub>ρ</sub> as a product:
+:<math>\displaystyle A_\rho(e^X) = e^{i\rho(X)} \prod_{\alpha>0}(1 - e^{-i\alpha(X)}),</math>
+where the product is over the positive roots.
+[[Weyl's dimension formula]] asserts that
+:<math>\displaystyle\chi_\lambda(1) \equiv {\rm dim}\, V = {\prod_{\alpha>0} (\lambda + \rho,\alpha)\over \prod_{\alpha>0} (\rho,\alpha)}.</math>
+where the [[inner product]] on <math>\mathfrak{t}^*</math> is that associated with the [[Killing form]] on <math>\mathfrak{k}</math>.
+Now
+*every irreducible representation of ''K'' extends  holomorphically to the complexification ''G''
+*every irreducible character χ<sub>λ</sub>(''k'') of ''K'' extends  holomorphically to the complexification  of ''K'' and <math>\mathfrak{t}^*</math>.
+* for every λ in Hom(''A'','''T''') = <math> i\mathfrak{t}^*</math>, there is a zonal spherical function φ<sub>λ</sub>.
+The '''Berezin–Harish–Chandra formula'''<ref name="harvnb|Helgason|1984"/> asserts that for ''X'' in <math>i\mathfrak{t}</math>
+:{| border="1" cellspacing="0" cellpadding="5"
+|<math> \varphi_\lambda(e^X) = {\chi_\lambda(e^X)\over\chi_\lambda(1)}.</math>
+|}
+In other words:
+*'''''the zonal spherical functions on a complex semisimple Lie group are given by analytic continuation of the formula for the normalised characters.'''''
+One of the simplest proofs<ref>{{harvnb|Helgason|1984|pp=432–433}}</ref> of this formula involves the ''radial component'' on ''A'' of the Laplacian on ''G'', a proof formally parallel to Helgason's reworking of [[Freudenthal]]'s classical proof of the [[Weyl character formula]], using the radial component on ''T'' of the Laplacian on ''K''.<ref>{{harvnb|Helgason|1984|pp= 501–502}}</ref>
+In the latter case the [[class function]]s on ''K'' can be identified with ''W''-invariant functions on ''T''. The
+radial component of Δ<sub>''K''</sub> on ''T'' is just the expression for the restriction of  Δ<sub>''K''</sub> to ''W''-invariant functions on ''T'', where
+it is given by the formula
+:<math>\displaystyle \Delta_K=  h^{-1}\circ \Delta_T \circ h + \|\rho\|^2,</math>
+where
+:<math>\displaystyle h(e^X) = A_\rho(e^X)</math>
+for ''X'' in <math>\mathfrak{t}</math>. If χ is a character with highest weight λ, it follows that φ = ''h''·χ satisfies
+:<math> \Delta_T \varphi= (\|\lambda + \rho\|^2 -\|\rho\|^2) \varphi.</math>
+Thus for every weight μ with non-zero [[Fourier coefficient]] in φ,
+:<math>\displaystyle \|\lambda +\rho\|^2 = \|\mu+\rho\|^2.</math>
+The classical argument of Freudenthal shows that μ + ρ must have the form ''s''(λ + ρ) for some ''s'' in ''W'', so the character formula
+follows from the antisymmetry of φ.
+Similarly ''K''-biinvariant functions on ''G'' can be identified with ''W''(''A'')-invariant functions on ''A''. The
+radial component of Δ<sub>''G''</sub> on ''A'' is just the expression for the restriction of  Δ<sub>''G''</sub> to ''W''(''A'')-invariant functions on ''A''.
+It is given by the formula
+:<math>\displaystyle \Delta_G=  H^{-1}\circ \Delta_A\circ H - \|\rho\|^2,</math>
+where
+:<math>\displaystyle H(e^X) = A_\rho(e^X)</math>
+for ''X'' in <math>i\mathfrak{t}</math>.
+The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function
+:<math>\displaystyle f= H\cdot\varphi,</math>
+which is an eigenfunction of the Laplacian Δ<sub>''A''</sub>. Since ''K'' is generated by copies of subgroups that are homomorphic images of SU(2) corresponding to [[root of a polynomial|simple roots]], its complexification ''G'' is generated by the corresponding homomorphic images of SL(2,'''C'''). The formula for zonal spherical functions of SL(2,'''C''') implies that ''f'' is a [[periodic function]] on  <math>i\mathfrak{t}</math> with respect to some [[lattice (discrete subgroup)|sublattice]]. Antisymmetry under the Weyl group and the argument of Freudenthal again imply that ψ must have the stated form up to a multiplicative constant, which can be determined using the Weyl dimension formula.
+==Example: SL(2,R)==
+{{See also|SL(2,R)|Representation theory of SL2(R)|Spectral theory of ordinary differential equations}}
+The theory of zonal spherical functions for [[SL(2,R)|SL(2,'''R''')]] originated in the work of Mehler in 1881 on hyperbolic geometry. He discovered the analogue of the Plancherel theorem, which was rediscovered by Fock in 1943. The corresponding eigenfunction expansion is termed the Mehler–Fock transform. It was already put on a firm footing in 1910 by [[Hermann Weyl]]'s important work on the [[spectral theory of ordinary differential equations]]. The radial part of the Laplacian in this case leads to a [[hypergeometric differential equation]], the theory of which was treated in detail by Weyl. Weyl's approach was subsequently generalised by Harish-Chandra to study zonal spherical functions and the corresponding Plancherel theorem for more general semimisimple Lie groups. Following the work of Dirac on the discrete series representations of SL(2,'''R'''), the general theory of unitary irreducible representations of  SL(2,'''R''') was developed independently by Bargmann, Harish-Chandra and Gelfand–Naimark. The irreducible representations of class one, or equivalently the theory of zonal spherical functions, form an important special case of this theory.
+The group ''G'' = [[SL(2,R)|SL(2,'''R''')]] is a [[Double covering group|double cover]] of the 3-dimensional [[Lorentz group]] SO(2,1), the [[symmetry group]] of the [[Hyperbolic space|hyperbolic plane]] with its [[Poincaré metric]]. It acts by [[Moebius transformation]]s. The upper half-plane can be identified with the unit disc by the [[Cayley transform]]. Under this identification ''G'' becomes identified with the group SU(1,1), also acting by Moebius transformations. Because the action is [[transitive action|transitive]], both spaces can be identified with ''G''/''K'', where ''K'' = [[SO(2)]]. The metric is invariant under ''G'' and the associated Laplacian is ''G''-invariant, coinciding with the image of the [[Casimir operator]]. In the upper half-plane model the Laplacian is given by the formula<ref name="harvnb|Helgason|1984" /><ref name="Lang1985" />
+:<math>\displaystyle\Delta=-4y^{2}(\partial_x^2 +\partial_y^2).</math>
+If ''s'' is a complex number and ''z'' = ''x + i y'' with ''y'' > 0, the function
+:<math>\displaystyle f_s(z) =y^{s}= \exp({s}\cdot\log y),</math>
+is an eigenfunction of Δ:
+:<math>\displaystyle \Delta f_s = 4s(1-s) f_s.</math>
+Since Δ commutes with ''G'', any left translate of ''f''<sub>''s''</sub> is also an eigenfunction with the same eigenvalue. In particular, averaging over ''K'', the function
+:{| border="1" cellspacing="0" cellpadding="5"
+|<math> \varphi_s(z) =\int_K f_s(k\cdot z)\, dk</math>
+|}
+is a ''K''-invariant eigenfunction of Δ on ''G''/''K''. When
+:<math>\displaystyle s={1\over 2} + i\tau,</math>
+with τ real, these functions give all the zonal spherical functions on ''G''. As with Harish-Chandra's more general formula for semisimple Lie groups, φ<sub>''s''</sub> is a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed by ''K'' in the [[principal series]]. Various arguments are available to prove that there are no others. One of the simplest classical [[Lie algebra]]ic arguments<ref name="harvnb|Helgason|1984"/><ref name="Lang1985" /><ref name="Bargmann 1947">{{harvnb|Bargmann|1947}}</ref><ref name="Howe 1992">{{harvnb|Howe | Tan |1992}}</ref><ref>{{harvnb|Wallach|1988}}</ref> is to note that, since Δ is an elliptic operator with analytic coefficients, by analytic elliptic regularity any eigenfunction is necessarily real analytic. Hence, if the zonal spherical function corresponds is the matrix coefficient for a vector ''v'' and representation σ, the vector ''v'' is an [[unitary representation|analytic vector]] for ''G'' and
+:<math>\displaystyle(\sigma(e^{X})v,v)= \sum_{n=0}^\infty (\sigma(X)^n v,v)/n!</math>
+for ''X'' in <math>i\mathfrak{t}</math>. The infinitesimal form of the irreducible unitary representations with a vector fixed by ''K'' were worked out classically by Bargmann.<ref name="Bargmann 1947"/><ref name="Howe 1992"/> They correspond precisely to the principal series of SL(2,'''R'''). It follows that the zonal spherical function corresponds to a principal series representation.
+Another classical argument<ref>{{harvnb|Helgason|2001|p= 405}}</ref> proceeds by showing that on radial functions the Laplacian has the form
+:<math>\displaystyle\Delta=-\partial_r^2 - \coth(r)\cdot \partial_r,</math>
+so that, as a function of ''r'', the zonal spherical function φ(''r'') must satisfy the [[ordinary differential equation]]
+:<math>\displaystyle\varphi^{\prime\prime} + \coth r\, \varphi^\prime = \alpha \, \varphi</math>
+for some constant α. The change of variables  ''t'' = sinh ''r''  transforms this equation into the [[hypergeometric differential equation]]. The general solution in terms of [[Legendre functions]] of complex index is given by<ref name="harvnb|Dieudonné |1978"/><ref>{{harvnb|Bateman|Erdelyi|1953|p=156}}</ref>
+:{| border="1" cellspacing="0" cellpadding="5"
+|<math>\varphi(r)=P_\rho(\cosh r) = {1\over 2\pi} \int_0^{2\pi} (\cosh r + \sinh r \, \cos \theta)^\rho \, d\theta,</math>
+|}
+where α = ρ(ρ+1). Further restrictions on ρ are imposed by boundedness and positive-definiteness of the zonal spherical function on ''G''.
+There is yet another approach, due to Mogens Flensted-Jensen, which derives the properties of the zonal spherical functions on SL(2,'''R'''), including the Plancherel formula, from the corresponding results for SL(2,'''C'''), which are simple consequences of the Plancherel formula and Fourier inversion formula for '''R'''. This "method of descent" works more generally, allowing results for a real semisimple Lie group to be derived by descent from the corresponding results for its complexification.<ref>{{harvnb|Flensted-Jensen|1978}}</ref><ref>{{harvnb|Helgason|1984|pages=489–491}}</ref>
+==Further directions==
+*''The theory of zonal functions that are not necessarily positive-definite.'' These are given by the same formulas as above, but without restrictions on the complex parameter ''s'' or ρ. They correspond to non-unitary representations.<ref name="harvnb|Helgason|1984" />
+*[[Plancherel theorem for spherical functions|''Harish-Chandra's eigenfunction expansion and inversion formula for spherical functions]].''<ref>{{harvnb|Helgason|1984|pp= 434–458}}</ref> This is an important special case of his [[Plancherel theorem]] for real semisimple Lie groups.
+*''The structure of the Hecke algebra''. Harish-Chandra and Godement proved that, as convolution algebras, there are natural isomorphisms between C<sub>c</sub><sup>∞</sup>(''K'' \ ''G'' / ''K'' ) and C<sub>c</sub><sup>∞</sup>(''A'')<sup>''W''</sup>, the subalgebra invariant under the Weyl group.<ref name="harvnb|Godement|1952"/> This is straightforward to establish for SL(2,'''R''').<ref name="Lang1985" />
+*''Spherical functions for [[Euclidean group|Euclidean motion groups]] and [[compact Lie group]]s''.<ref name="harvnb|Helgason|1984"/>
+*''Spherical functions for [[p-adic]] Lie groups''. These were studied in depth by Satake and [[Ian G. Macdonald|Macdonald]].<ref>{{harvnb|Satake|1963}}</ref><ref>{{harvnb|Macdonald|1971}}</ref> Their study, and that of the associated Hecke algebras, was one of the first steps in the extensive representation theory of semisimple p-adic Lie groups, a key element in the [[Langlands program]].
+==See also==
+*[[Plancherel theorem for spherical functions]]
+*[[Hecke algebra#Hecke algebra of a locally compact group|Hecke algebra of a locally compact group]]
+*[[Representations of Lie groups]]
+*[[Non-commutative harmonic analysis]]
+*[[Tempered representation]]
+*[[Positive definite function on a group]]
+*[[Symmetric space]]
+*[[Gelfand pair]]
+==Notes==
+{{reflist|3}}
+==References==
+*{{citation|first=V.|last= Bargmann|authorlink=Valentine Bargmann|title=Irreducible Unitary Representations of the Lorentz Group|journal= Annals of Mathematics|volume= 48|pages= 568–640|year=1947|doi=10.2307/1969129|issue=3|publisher=Annals of Mathematics|jstor=1969129}}
+*{{citation|last=Barnett|first=Adam|last2= Smart|first2=Nigel P.|title=Mental Poker revisited, in Cryptography and Coding|pages= 370–383|year=2003|journal= [[Lecture Notes in Computer Science]]|volume= 2898|publisher= Springer-Verlag}}
+*{{citation|title=Higher transcendental functions, Vol.I|first= Harry|last= Bateman|authorlink1=Harry Bateman|first2= Arthur|last2= Erdélyi|authorlink2=Arthur Erdélyi|year= 1953
+|publisher=McGraw–Hill|ISBN= 0-07-019546-3|url=http://apps.nrbook.com/bateman/Vol1.pdf}}
+*{{Citation|last=Berezin|first= F. A.|authorlink=Felix Berezin|title=Операторы Лапласа на полупростых группах|trans_title= Laplace operators on semisimple groups|journal=[[Doklady Akademii Nauk SSSR]] |volume=  107  |year=1956|pages =9–12|postscript=.}}
+*{{citation|last=Berezin|first= F. A.|title= Representation of complex semisimple Lie groups in Banach space|journal=Doklady Akademii Nauk SSSR |volume=  110  |year=1956b|pages =897–900}}
+*{{springer|id=s/s086670|title=Spherical functions|first1=Yu.A.|last1=Brychkov|first2=A.P.|last2=Prudnikov|year=2001}}
+*{{citation|first=Pierre|last=Cartier|authorlink=Pierre Cartier (mathematician)|title=Structure topologique des groupes de Lie généraux, Exposé No. 22|series=Séminaire "Sophus Lie"|volume=1|year=1954–1955 |url=http://archive.numdam.org/article/SSL_1954-1955__1__A24_0.pdf}}.
+*{{citation|title=Heat Kernels and Spectral Theory|first=E. B.|last= Davies|year=1990|publisher=Cambridge University Press|ISBN= 0-521-40997-7}}
+*{{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis, Vol. V|publisher=Academic Press|year=1977}}
+*{{citation|first=Jean|last=Dieudonné|authorlink=Jean Dieudonné|title=Treatise on Analysis, Vol. VI|publisher=Academic Press|year=1978|isbn=0-12-215506-8}}
+*{{citation|last=Dirac|first= P. A. M.|authorlink=P. A. M. Dirac|title=Unitary representations of the Lorentz group|journal=[[Proceedings of the Royal Society A]]| volume=183|year=1945|pages= 284–295|doi=10.1098/rspa.1945.0003|issue=994}}
+*{{citation|last=Dixmier| first=Jacques| authorlink=Jacques Dixmier|title=Les algèbres d'opérateurs dans l'espace hilbertien (algèbres de von Neumann) |series=Les Grands Classiques Gauthier-Villars.|publisher= Éditions Jacques Gabay|year= 1996|ISBN= 2-87647-012-8}}
+*{{citation|last=Flensted-Jensen|first=Mogens|title=Spherical functions of a real semisimple Lie group. A method of reduction to the complex case
+|journal=J. Funct. Anal.|volume= 30|year=1978|pages=106–146|doi=10.1016/0022-1236(78)90058-7}}
+*{{citation|last1=Gelfand|first1=I.M.|authorlink1=Israel Gelfand|last2=Naimark|first2=M.A.|authorlink2=Mark Naimark|title=Unitary representations of the Lorentz group|journal=Izv. Akad. Nauk SSSR, Ser. Mat.|year=1947|volume=37|pages=411–504}}
+*{{citation|last1=Gelfand|first1=I.M.|authorlink1=Israel Gelfand|last2=Naimark|first2=M.A.|authorlink2=Mark Naimark|title=An analogue of Plancherel's theorem for the complex unimodular group|journal=Doklady Akademii Nauk SSSR| year =1948|volume=63|pages=609–612}}
+*{{citation|last1=Gelfand|first1=I.M.|authorlink1=Israel Gelfand|last2=Naimark|first2=M.A.|authorlink2=Mark Naimark|title=Unitary representations of the unimodular group containing the identity representation of the unitary subgroup|year=1952|volume =1|pages=423–475|journal=Trudy Moscov. Mat. Obšč.}}
+*{{citation|first=Roger|last= Godement|authorlink=Roger Godement|title=A theory of spherical functions. I|journal=Transactions of the American Mathematical Society|year=1952|volume=73|pages=496–556|doi=10.2307/1990805|issue=3|publisher=American Mathematical Society|jstor=1990805}}
+*{{citation|last2=Wallach|first2=Nolan|last=Goodman|first=Roe|title=Representations and Invariants of the Classical Groups|publisher=Cambridge University Press|year=1998|ISBN= 0-521-66348-2}}
+*{{citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Infinite irreducible representations of the Lorentz group|journal=[[Proceedings of the Royal Society A]]|volume= 189| year=1947|pages= 372–401|doi=10.1098/rspa.1947.0047|issue=1018}}
+*{{citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Representations of Semisimple Lie Groups. II|journal=Trans. Amer. Math. Soc.|year=1954a|volume=76|pages=26–65|doi=10.2307/1990743}}
+*{{citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Representations of Semisimple Lie Groups. III|journal=Trans. Amer. Math. Soc.|year=1954b|volume=76|pages=234–253|doi=10.2307/1990767|issue=2|publisher=American Mathematical Society|jstor=1990767}} (Simplification of formula for complex semisimple Lie groups)
+*{{citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=The Plancherel formula for complex semisimple Lie groups|journal=Trans. Amer. Math. Soc.|year=1954c|volume=76|pages=485–528|doi=10.2307/1990793|issue=3|publisher=American Mathematical Society|jstor=1990793}} (Second proof of formula for complex semisimple Lie groups)
+*{{citation|last=Harish-Chandra|authorlink=Harish-Chandra|title=Spherical functions on a semisimple Lie group I, II|journal=Amer. J. Math.|year=1958|volume=80|pages=241–310, 553–613|doi=10.2307/2372786|issue=2|publisher=The Johns Hopkins University Press|jstor=2372786}} (Determination of Plancherel measure)
+*{{citation|first=Sigurdur|last=Helgason|authorlink=Sigurdur Helgason (mathematician)|title=Differential geometry and symmetric spaces (reprint of 1962 edition)|year=2001|publisher=American Mathematical Society|ISBN= 0-8218-2735-9}}
+*{{citation|first=Sigurdur|last=Helgason|authorlink=Sigurdur Helgason (mathematician)|title=Differential geometry, Lie groups and symmetric spaces|year=1978|publisher=Academic Press|ISBN= 0-12-338460-5}}
+*{{citation|first=Sigurdur|last=Helgason|authorlink=Sigurdur Helgason (mathematician)|title=Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions|year=1984|publisher=Academic Press|ISBN= 0-12-338301-3}}
+*{{citation|last=Hochschild|first = Gerhard P.|title= The structure of Lie groups|publisher=Holden–Day|year=1965}}
+*{{citation|title=Non-abelian Harmonic Analysis: Applications of SL(2,'''R''')|first=Roger|last= Howe|first2= Eng-chye|last2= Tan|
+year=1992|publisher=Springer-Verlag|series=Universitext|ISBN= 0-387-97768-6}}
+*{{citation|last= Korányi|first= Adam|title=Analysis on symmetric cones|series=Oxford Mathematical Monographs|publisher=Oxford University Press|year= 1994|ISBN= 0-19-853477-9}}, Chapter XIV.
+*{{citation|last=Kostant|first= Bertram|authorlink=Bertram Kostant|title=On the existence and irreducibility of certain series of representations|journal=Bull. Amer. Math. Soc.|volume= 75 |year=1969|pages =627–642|doi=10.1090/S0002-9904-1969-12235-4|issue=4}}
+*{{citation|last=Kunze|first=Raymond A.|last2 =Stein|first2= Elias M.|authorlink2=Elias Stein|title=Analytic continuation of the principal series |volume=67|year= 1961|pages=593–596| journal=[[Bull. Amer. Math. Soc.]]|doi=10.1090/S0002-9904-1961-10705-2|issue=6}}
+*{{citation|first=Serge|last=Lang|authorlink=Serge Lang|title=SL(2,'''R''')|publisher=Springer-Verlag|series=Graduate Texts in Mathematics|volume=105|
+year=1985|ISBN= 0-387-96198-4}}
+*{{citation|first=Ian G.|last=Macdonald|authorlink=Ian G. Macdonald|title=Spherical Functions on a Group of p-adic Type|series=Publ. Ramanujan Institute|volume=2|year=1971|publisher=University of Madras}}
+*{{citation|last=Stein|first=Elias M.|authorlink=Elias Stein|title=Analytic continuation of group representations|journal=Advances in Math.|volume= 4 |year=1970|pages= 172–207|doi=10.1016/0001-8708(70)90022-8|issue=2}}
+*{{citation|first=I.|last=Satake|title=Theory of spherical functions on reductive algebraic groups over p-adic fields|journal=Publ. Math. I.H.E.S.|year=1963|volume=18|pages=5–70}}
+*{{citation|first=Nolan|last=Wallach|title=Harmonic Analysis on Homogeneous Spaces|year=1973|publisher=Marcel Decker|ISBN= 0-8247-6010-7}}
+*{{citation|first=Nolan|last=Wallach|title=Real Reductive Groups I|year=1988|publisher=Academic Press|ISBN= 0-12-732960-9}}
+*{{citation|last=Takahashi|first= R.|title=Sur les représentations unitaires des groupes de Lorentz généralisés|journal=Bull. Soc. Math. France|
+volume= 91| year=1963|pages= 289–433}}
+==External links==
+*{{citation|last=Casselman|first=William|title=Notes on spherical functions|url=http://www.math.ubc.ca/~cass/research.html}}
+{{DEFAULTSORT:Zonal Spherical Function}}
+[[Category:Representation theory of Lie groups]]
+[[Category:Harmonic analysis]]
+[[Category:Functional analysis]]
+[[Category:Types of functions]]

Birnbaum–Orlicz space: Difference between revisions

Revision as of 19:08, 7 January 2014

Definitions

Properties

Gelfand pairs

Cartan–Helgason theorem

Harish-Chandra's formula

Eigenfunctions

Example: SL(2,C)

Complex case

Example: SL(2,R)

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Notes

References

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