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 subgroup In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semi ...

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 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

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 In mathematics, a K-finite function is a type of generalized trigonometric polynomial. Here ''K'' is some compact group, and the generalization is from the circle group ''T''. From an abstract point of view, the characterization of trigonometri ...

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 In mathematics, particularly in functional analysis and ring theory, an approximate identity is a net in a Banach algebra or ring (generally without an identity) that acts as a substitute for an identity element. Definition A right approximate ...

, and the approximately unital R(''g'',''K'')- modules are the same as (''g'',''K'') modules.

Definition

The Zuckerman functor Γ is defined by :

\Gamma^_(W) = \hom_(R(g,K),W)_K

and the Bernstein functor Π is defined by :

\Pi^_(W) = R(g,K)\otimes_W.

References

* David A. Vogan, ''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'',
preface
http://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 The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent scholar ...

, 1978. Representation theory Functors