Zuckerman functor

This article is about the Zuckerman induction functor, which is not the same as the (Zuckerman) translation 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.
* 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.
* WK is the subspace of K-finite vectors of a representation W of K.
* 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.
* 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

Γg,L∩Kg,K(W)=homR(g,L∩K)(R(g,K),W)K

and the Bernstein functor Π is defined by

Πg,L∩Kg,K(W)=R(g,K)⊗R(g,L∩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 Barbasch MR1330919
* 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* Gregg J. Zuckerman, Construction of representations via derived functors, unpublished lecture series at the Institute for Advanced Study, 1978.

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