|
|
| Line 1: |
Line 1: |
| {{dablink|This article is about the Zuckerman induction functor, which is not the same as the (Zuckerman) [[translation functor]].}}
| | Hi there, I am Yoshiko Villareal but I never truly favored that title. The job he's been occupying for years is a messenger. Arizona has always been my living place but my spouse wants us to move. To perform croquet is something that I've done for years.<br><br>my blog :: extended auto warranty - [http://wp.shizzlaz.eu/index.php?mod=users&action=view&id=23474 Recommended Web site], |
| | |
| In [[mathematics]], a '''Zuckerman functor''' is used to construct representations of real [[Reductive group|reductive]] [[Lie group]]s from representations of [[Levi subgroup]]s. They were introduced by [[Gregg Zuckerman]] (1978). The '''Bernstein functor''' is closely related.
| |
| ==Notation and terminology==
| |
| *''G'' is a connected reductive real affine [[algebraic group]] (for simplicity; the theory works for more general groups), and ''g'' is the [[Lie algebra]] of ''G''. ''K'' is a [[maximal compact subgroup]] of ''G''.
| |
| *''L'' is a [[Levi subgroup]] of ''G'', the centralizer of a compact connected abelian subgroup, and *''l'' is the Lie algebra of ''L''.
| |
| *A representation of ''K'' is called '''[[K-finite]]''' if every vector is contained in a finite dimensional representation of ''K''. Denote by ''W''<sub>''K''</sub> the subspace of ''K''-finite vectors of a representation ''W'' of ''K''.
| |
| *A '''(g,K)-module''' is a vector space with compatible actions of ''g'' and ''K'', on which the action of ''K'' is ''K''-finite.
| |
| *R(''g'',''K'') is the [[Hecke algebra]] of ''G'' of all distributions on ''G'' with support in ''K'' that are left and right ''K'' finite. This is a ring which does not have an identity but has an [[approximate identity]], and the approximately unital R(''g'',''K'')- modules are the same as (''g'',''K'') modules.
| |
| | |
| ==Definition==
| |
| The Zuckerman functor Γ is defined by
| |
| | |
| :<math>\Gamma^{g,K}_{g,L\cap K}(W) = \hom_{R(g,L\cap K)}(R(g,K),W)_K</math>
| |
| | |
| and the Bernstein functor Π is defined by
| |
| | |
| :<math>\Pi^{g,K}_{g,L\cap K}(W) = R(g,K)\otimes_{R(g,L\cap K)}W.</math> | |
| | |
| ==Applications==
| |
| {{Empty section|date=July 2010}}
| |
| ==References==
| |
| *[[David A. Vogan]], ''Representations of real reductive Lie groups'', ISBN 3-7643-3037-6
| |
| *[[A. Knapp]], David A. Vogan, ''Cohomological induction and unitary representations'', ISBN 0-691-03756-6 [http://www.math.sunysb.edu/~aknapp/books/brown/kv-preface.pdf preface][http://www.ams.org/bull/1999-36-03/S0273-0979-99-00782-X/S0273-0979-99-00782-X.pdf review by D. Barbasch]{{MathSciNet|id=1330919}}
| |
| *David A. Vogan ''Unitary Representations of Reductive Lie Groups.'' (AM-118) (Annals of Mathematics Studies) ISBN 0-691-08482-3
| |
| *G. J. Zuckerman, ''Construction of representations via derived functors'', unpublished lecture series at the I. A. S., 1978.
| |
| | |
| [[Category:Representation theory]]
| |
| [[Category:Functors]]
| |
Hi there, I am Yoshiko Villareal but I never truly favored that title. The job he's been occupying for years is a messenger. Arizona has always been my living place but my spouse wants us to move. To perform croquet is something that I've done for years.
my blog :: extended auto warranty - Recommended Web site,