In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Zuckerman functor is used to construct representations of real
reductive Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s from representations of
Levi subgroups. 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
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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''
''K'' 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
:
and the Bernstein functor Π is defined by
:
References
*
David A. Vogan
David Alexander Vogan, Jr. (born September 8, 1954) is a mathematician at the Massachusetts Institute of Technology who works on unitary representations of simple Lie groups.
While studying at the University of Chicago, he became a Putnam Fe ...
, ''Representations of real reductive Lie groups'',
*
Anthony W. Knapp
Anthony W. Knapp (born 2 December 1941, Morristown, New Jersey) is an American mathematician at the State University of New York, Stony Brook working on representation theory, who classified the tempered representations of a semisimple Lie grou ...
, David A. Vogan, ''Cohomological induction and unitary representations'',
prefacehttp://www.ams.org/bull/1999-36-03/S0273-0979-99-00782-X/S0273-0979-99-00782-X.pdf review by Dan Barbasch]
*David A. Vogan, ''Unitary Representations of Reductive Lie Groups.'' (AM-118) (Annals of Mathematics Studies) {{isbn, 0-691-08482-3
*
Gregg Zuckerman, Gregg J. Zuckerman, ''Construction of representations via derived functors'', unpublished lecture series at the
Institute for Advanced Study, 1978.
Representation theory
Functors