# Zuckerman functor

In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups 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 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 WK 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

$\Gamma _{g,L\cap K}^{g,K}(W)=\hom _{R(g,L\cap K)}(R(g,K),W)_{K}$ and the Bernstein functor Π is defined by

$\Pi _{g,L\cap K}^{g,K}(W)=R(g,K)\otimes _{R(g,L\cap K)}W.$ This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.