# 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

${\displaystyle \Gamma _{g,L\cap K}^{g,K}(W)=\hom _{R(g,L\cap K)}(R(g,K),W)_{K}}$

and the Bernstein functor Π is defined by

${\displaystyle \Pi _{g,L\cap K}^{g,K}(W)=R(g,K)\otimes _{R(g,L\cap K)}W.}$

## References

• David A. Vogan, Representations of real reductive Lie groups, ISBN 3-7643-3037-6
• Anthony W. Knapp, David A. Vogan, Cohomological induction and unitary representations, ISBN 0-691-03756-6 prefacereview by Dan BarbaschMR1330919
• David A. Vogan, Unitary Representations of Reductive Lie Groups. (AM-118) (Annals of Mathematics Studies) ISBN 0-691-08482-3
• Gregg J. Zuckerman, Construction of representations via derived functors, unpublished lecture series at the Institute for Advanced Studies, 1978.