Holding value

In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).

Structure of the complexified Lie algebra

We choose a basis H, X, Y for the complexification of the Lie algebra of SL(2,R) so that iH generates the Lie algebra of a compact Cartan subgroup K (so in particular unitary representations split as a sum of eigenspaces of H), and {H,X,Y} is an sl₂-triple, which means that they satisfy the relations

$[H,X]=2X,[H,Y]=-2Y,[X,Y]=H.$

One way of doing this is as follows:

$H=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right)$ corresponding to the subgroup K of matrices $\left(\begin{array}{cc}\cos (\theta )& -\sin (\theta )\\ \sin (\theta )& \cos (\theta )\end{array}\right)$

$X=\frac{1}{2}\left(\begin{array}{cc}1& i\\ i& -1\end{array}\right)$

$Y=\frac{1}{2}\left(\begin{array}{cc}1& -i\\ -i& -1\end{array}\right)$

The Casimir operator Ω is defined to be

$\mathrm{\Omega }=H^2+1+2XY+2YX.$

It generates the center of the universal enveloping algebra of the complexified Lie algebra of SL(2,R). The Casimir element acts on any irreducible representation as multiplication by some complex scalar μ². Thus in the case of the Lie algebra sl₂, the infinitesimal character of an irreducible representation is specified by one complex number.

The center Z of the group SL(2,R) is a cyclic group {I,-I} of order 2, consisting of the identity matrix and its negative. On any irreducible representation, the center either acts trivially, or by the nontrivial character of Z, which represents the matrix -I by multiplication by -1 in the representation space. Correspondingly, one speaks of the trivial or nontrivial central character.

The central character and the infinitesimal character of an irreducible representation of any reductive Lie group are important invariants of the representation. In the case of irreducible admissible representations of SL(2,R), it turns out that, generically, there is exactly one representation, up to an isomorphism, with the specified central and infinitesimal characters. In the exceptional cases there are two or three representations with the prescribed parameters, all of which have been determined.

Finite dimensional representations

For each nonnegative integer n, the group SL(2,R) has an irreducible representation of dimension n+1, which is unique up to an isomorphism. This representation can be constructed in the space of homogeneous polynomials of degree n in two variables. The case n=0 corresponds to the trivial representation. An irreducible finite dimensional representation of a noncompact simple Lie group of dimension greater than 1 is never unitary. Thus this construction produces only one unitary representation of SL(2,R), the trivial representation.

The finite-dimensional representation theory of the noncompact group SL(2,R) is equivalent to the representation theory of SU(2), its compact form, essentially because their Lie algebras have the same complexification and they are "algebraically simply connected". (More precisely the group SU(2) is simply connected and SL(2,R) is not, but has no non-trivial algebraic central extensions.) However, in the general infinite-dimensional case, there is no close correspondence between representations of a group and the representations of its Lie algebra. In fact, it follows from the Peter-Weyl theorem that all irreducible representations of the compact Lie group SU(2) are finite-dimensional and unitary. The situation with SL(2,R) is completely different: it possesses infinite-dimensional irreducible representations, some of which are unitary, and some are not.

Principal series representations

A major technique of constructing representations of a reductive Lie group is the method of parabolic induction. In the case of the group SL(2,R), there is up to conjugacy only one proper parabolic subgroup, the Borel subgroup of the upper-triangular matrices of determinant 1. The inducing parameter of an induced principal series representation is a (possibly non-unitrary) character of the multiplicative group of real numbers, which is specified by choosing ε = ± 1 and a complex number μ. The corresponding principal series representation is denoted I_ε,μ. It turns out that ε is the central character of the induced representation and the complex number μ may be identified with the infinitesimal character via the Harish-Chandra isomorphism.

The principal series representation I_ε,μ (or more precisely its Harish-Chandra module of K-finite elements) admits a basis consisting of elements w_j, where the index j runs through the even integers if ε=1 and the odd integers if ε=-1. The action of X, Y, and H is given by the formulas

$H(w_j)=j{w}_j$

$X(w_j)=\frac{\mu +j+1}{2}{w}_{j+2}$

$Y(w_j)=\frac{\mu -j+1}{2}{w}_{j-2}$

Admissible representations

Using the fact that it is an eigenvector of the Casimir operator and has an eigenvector for H, it follows easily that any irreducible admissible representation is a subrepresentation of a parabolically induced representation. (This also is true for more general reductive Lie groups and is known as Casselman's subrepresentation theorem.) Thus the irreducible admissible representations of SL(2,R) can be found by decomposing the principal series representations I_ε,μ into irreducible components and determining the isomorphisms. We summarize the decompositions as follows:

• I_ε,μ is reducible if and only if μ is an integer and ε=−(−1)^μ. If I_ε,μ is irreducible then it is isomorphic to I_ε,−μ.
• I_{−1, 0} splits as the direct sum I_ε,0 = D₊₀ + D₋₀ of two irreducible representations, called limit of discrete series representations. D₊₀ has a basis w_j for j≥1, and D_-0 has a basis w_j for j≤−1,
• If I_ε,μ is reducible with μ>0 (so ε=−(−1)^μ) then it has a unique irreducible quotient which has finite dimension μ, and the kernel is the sum of two discrete series representations D_+μ + D_−μ. The representation D_μ has a basis w_μ+j for j≥1, and D_-μ has a basis w_−μ−j for j≤−1.
• If I_ε,μ is reducible with μ<0 (so ε=−(−1)^μ) then it has a unique irreducible subrepresentation, which has finite dimension μ, and the quotient is the sum of two discrete series representations D_+μ + D_−μ.

This gives the following list of irreducible admissible representations:

• A finite dimensional representation of dimension μ for each positive integer μ, with central character −(−1)^μ.
• Two limit of discrete series representations D₊₀, D₋₀, with μ=0 and non-trivial central character.
• Discrete series representations D_μ for μ a non-zero integer, with central character −(−1)^μ.To succeed in selling a home, it is advisable be competent in real estate advertising and marketing, authorized, monetary, operational aspects, and other information and skills. This is essential as a result of you want to negotiate with more and more sophisticated buyers. You could outperform rivals, use latest technologies, and stay ahead of the fast altering market.
  
  Home is where the center is, and choosing the right house is a part of guaranteeing a contented expertise in Singapore. Most expats sign up for a two-year lease with the option to resume, so it is value taking the time to choose a neighbourhood that has the services you want. The experts at Expat Realtor have compiled the next data that will help you negotiate your means by way of the property minefield. Some government state properties for rent. Over 2000 units available for lease however occupancy is often excessive. Some properties come under a bidding system. Their property brokers embody DTZ and United Premas. Up to date serviced residences located just off Orchard Highway. one hundred sixty Orchard Highway, #06-01 Orchard Level, Singapore 238842. Institute Of Property Agents
  
  There is no such thing as a deal too small. Property agents who're willing to find time for any deal even when the commission is small are those you want in your side. They also show humbleness and might relate with the average Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even when they get rejected a hundred times, they still come back for more. These are the property brokers who will find consumers what they want finally, and who would be the most profitable in what they do. four. Honesty and Integrity
  
  As a realtor, you're our own business. Due to this fact, it is imperative that you handle yours prices and spend money correctly in order to market your property successfully. Also, beware of mentors who always ask you to pay for pointless costs. Such mentors typically are recruiting to develop a staff and see you as a option to defray advertising and marketing prices. For foreigners who want to register with CEA as salespersons, they might want to have a valid Employment Cross (EP) issued by the Ministry of Manpower (MOM). They should consult an property agent that is ready to assist their future registration software, who would then examine with CEA. Thereafter, after they register for the RES Course, they might want to produce a letter of assist from the property agent."
  
  Main Real Property Brokers with in depth local knowledge, Carole Ann, Elizabeth and their group of extremely skilled property consultants provide a personalised service, for those looking to buy, lease or promote in Singapore. Relocation companies out there. Properties for the aesthete. Boutique real property agency for architecturally distinguished, unique properties for rent and on the market. Caters to the niche market of design-savvy people. Sale, letting and property management and taxation services. three Shenton Means, #10-08 Shenton Home, Singapore 068805. Buy property, promote or leasing estate company. 430 Lorong 6 Toa Payoh, #08-01 OrangeTee Constructing, Singapore 319402. HIGH Date / Age of property Estate Agents and Home Search Services Property Information Highlights Prime Achievers
  
  From the above info, you may see that saving on agent's commission will not cover the expenses wanted to market your home efficiently. As well as, it's essential make investments a whole lot of time, vitality and effort. By taking yourself away from your work and other endeavors, additionally, you will incur unnecessary opportunity prices. There may be additionally no assurance you could beat the market and get the outcomes you need. That is why you want an agent - not simply an ordinary agent - you want knowledgeable and competent specialist, geared up with the best instruments and knowledge to serve you and lead you to success! Within the midst of this ‘uniquely Singapore' Property GSS, our most needed foreign customers are nowhere to be seen. Different types of Public Residential properties
  
  Based on Kelvin, other agents may also make use of your agent's listings. "If your pricing is on the excessive aspect, these brokers may use your house to persuade their patrons why Http://Trafficstooges.Com/Singapore-Property-Condominium they should purchase another residence." To counter this, Kelvin says it is crucial for your agent to supply a current market analysis before putting up your private home for sale. "This helps you worth your property appropriately and realistically." When property is made accessible (HIGH is issued) to the client. Becoming a successful property agent is a distinct story altogether! Hi, I would like to ask how I might be a property agent and whether there are courses I might take. And if I need to be at a certain age. www. Property BUYER com.sg (your impartial Mortgage Advisor) In private properties in
• Two families of irreducible principal series representations I_ε,μ for ε≠−(−1)^μ (where I_ε,μ is isomorphic to I_ε,−μ).

Relation with the Langlands classification

According to the Langlands classification, the irreducible admissible representations are parametrized by certain tempered representations of Levi subgroups M of parabolic subgroups P=MAN. This works as follows:

• The discrete series, limit of discrete series, and unitary principal series representations I_ε,μ with μ imaginary are already tempered, so in these cases the parabolic subgroup P is SL(2,R) itself.
• The finite dimensional representations and the representations I_ε,μ for ℜμ>0, μ not an integer or ε≠−(−1)^μ are the irreducible quotients of the principal series representations I_ε,μ for ℜμ>0, which are induced from tempered representations of the parabolic subgroup P=MAN of upper triangular matrices, with A the positive diagonal matrices and M the center of order 2. For μ a positive integer and ε=−(−1)^μ the principal series representation has a finite dimensional representation as its irreducible quotient, and otherwise it is already irreducible.

Unitary representations

The irreducible unitary representations can be found by checking which of the irreducible admissible representations admit an invariant positively-definite Hermitian form. This results in the following list of unitary representations of SL(2,R):

• The trivial representation (the only finite-dimensional representation in this list).
• The two limit of discrete series representations D₊₀, D₋₀.
• The discrete series representations D_k, indexed by non-zero integers k. They are all distinct.
• The two families of irreducible principal series representation, consisting of the spherical principal series I_+,iμ indexed by the real numbers μ, and the non-spherical unitary principal series I_-,iμ indexed by the non-zero real numbers μ. The representation with parameter μ is isomorphic to the one with parameter −μ, and there are no further isomorphisms between them.
• The complementary series representations I_+,μ for 0<|μ|<1. The representation with parameter μ is isomorphic to the one with parameter −μ, and there are no further isomorphisms between them.

Of these, the two limit of discrete series representations, the discrete series representations, and the two families of principal series representations are tempered, while the trivial and complementary series representations are not tempered.

References

• V. Bargmann, Irreducible Unitary Representations of the Lorentz Group, The Annals of Mathematics, 2nd Ser., Vol. 48, No. 3 (Jul., 1947), pp. 568–640
• Gelfand, I.; Neumark, M. Unitary representations of the Lorentz group. Acad. Sci. USSR. J. Phys. 10, (1946), pp. 93–94
• Harish-Chandra, Plancherel formula for the 2×2 real unimodular group. Proc. Nat. Acad. Sci. U.S.A. 38 (1952), pp. 337–342
• Roger Howe, Eng-Chye Tan, Nonabelian harmonic analysis. Applications of SL(2,R). Universitext. Springer-Verlag, New York, 1992. ISBN 0-387-97768-6
• Knapp, Anthony W. Representation theory of semisimple groups. An overview based on examples. Reprint of the 1986 original. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 2001. xx+773 pp. ISBN 0-691-09089-0
• Kunze, R. A.; Stein, E. M. Uniformly bounded representations and harmonic analysis of the 2×2 real unimodular group. Amer. J. Math. 82 (1960), pp. 1–62
• D. Vogan, Representations of real reductive Lie groups, ISBN 3-7643-3037-6
• N. R. Wallach, Real reductive groups I. ISBN 0-12-732960-9

Minicourse

The videos of the SL(2,R) Summer School in Utah in June 2006 provides a great introduction on master level: Homepage of Utah Summer School 2006.

See also

• spin (physics)
• Representation theory of SU(2)