In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a Zuckerman functor is used to construct representations of real
reductive Lie group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.
A manifold is a space that locally resembles Eucli ...
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
In mathematics, an algebraic group is an algebraic variety endowed with a group structure that is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory.
Man ...
(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 operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
of ''G''.
*''K'' is a
maximal compact subgroup
In mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. T ...
of ''G''.
*A (g,K)-module is a vector space with compatible actions of ''g'' and ''K'', on which the action of ''K'' is ''K''-finite. A representation of ''K'' is called
K-finite if every vector is contained in a finite-dimensional representation of ''K''.
*''W''
''K'' is the subspace of ''K''-finite vectors of a representation ''W'' of ''K''.
*R(''g'',''K'') is the
Hecke algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators, which are named after Erich Hecke.
Properties
The algebra is a commutative ring.
In the classical elliptic modular form theory, the Hecke operators ''T'n'' with ' ...
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.
*''L'' is a
Levi subgroup of ''G'', the centralizer of a compact connected abelian subgroup, and ''l'' is the Lie algebra of ''L''.
Definition
The Zuckerman functor Γ is defined by
:
and the Bernstein functor Π is defined by
:
References
*
David A. Vogan, ''Representations of real reductive Lie groups'',
*
Anthony W. Knapp, 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)
*
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) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
, 1978.
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Representation theory
Functors