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

## 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*, 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*Construction of representations via derived functors*, unpublished lecture series at the Institute for Advanced Studies, 1978.

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