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2013-03-01T09:27:45Z

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	~~Alyson~~ is ~~what my husband loves~~ to ~~call me but I don~~'~~t like when individuals use my full name~~. ~~Office supervising is my profession~~. ~~She is truly fond~~ of ~~caving but she doesn~~'~~t have~~ the ~~time lately~~. I'~~ve always loved living~~ in ~~Mississippi~~.<br><br>~~my blog:~~ [http://~~formalarmour~~.~~com~~/~~index~~.~~php?do=~~/~~profile~~-~~26947~~/~~info/ free psychic~~]		In [[mathematics]], '''parabolic induction ''' is a method of constructing [[group representation\|representation]]s of a [[reductive group]] from representations of its [[parabolic subgroup]]s.

			If ''G'' is a reductive algebraic group and <math>P=MAN</math> is the [[Langlands decomposition]] of a parabolic subgroup ''P'', then parabolic induction consists of taking a representation of <math>MA</math>, extending it to ''P'' by letting ''N'' act trivially, and [[induced representation\|inducing]] the result from ''P'' to ''G''.

			There are some generalizations of parabolic induction using [[cohomology]], such as [[cohomological parabolic induction]] and [[Deligne–Lusztig theory]].

			==Philosophy of cusp forms==

			The ''philosophy of [[cusp form]]s'' was a slogan of [[Harish-Chandra]], expressing his idea of a kind of reverse engineering of [[automorphic form]] theory, from the point of view of [[representation theory]].<ref>Daniel Bump, ''Automorphic Forms and Representations'' (1998), p. 421.</ref> The [[discrete group]] Γ fundamental to the classical theory disappears, superficially. What remains is the basic idea that representations in general are to be constructed by parabolic induction of [[cuspidal representation]]s.<ref>See Daniel Bump, ''Lie Groups'' (2004), p. 397.</ref> A similar philosophy was enunciated by [[Israel Gelfand]],<ref>{{Citation \|first=I. M. \|last=Gelfand \|title=Automorphic functions and the theory of representations \|work=Proceedings, International Congress of Mathematicians \|location=Stockholm \|year=1962 \|pages=74–85 }}.</ref> and the philosophy is a precursor of the [[Langlands program]]. A consequence for thinking about representation theory is that [[cuspidal representation]]s are the fundamental class of objects, from which other representations may be constructed by procedures of induction.

			According to [[Nolan Wallach]]<ref>[http://www.math.ucsd.edu/~nwallach/luminy-port2.pdf PDF], p.80.</ref>

			<blockquote>Put in the simplest terms the "philosophy of cusp forms" says that for each Γ-conjugacy classes of Q-rational parabolic subgroups one should construct automorphic functions (from objects from spaces of lower dimensions) whose constant terms are zero for other conjugacy classes and the constant terms for [an]<!-- source says "and" --> element of the given class give all constant terms for this parabolic subgroup. This is almost possible and leads to a description of all automorphic forms in terms of these constructs and cusp forms. The construction that does this is the [[Eisenstein series]].</blockquote>

			==Notes==
			{{Reflist}}

			==References==
			*A. W. Knapp, ''Representation Theory of Semisimple Groups: An Overview Based on Examples'', Princeton Landmarks in Mathematics, Princeton University Press, 2001. ISBN 0-691-09089-0.

			[[Category:Representation theory]]

			{{algebra-stub}}

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Alyson is what my husband loves to call me but I don't like when individuals use my full name. Office supervising is my profession. She is truly fond of caving but she doesn't have the time lately. I've always loved living in Mississippi.<br><br>my blog: [http://formalarmour.com/index.php?do=/profile-26947/info/ free psychic]

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