Zuckerman functor

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.
• W**K 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

:\Gamma^{g,K}{g,L\cap K}(W) = \hom{R(g,L\cap K)}(R(g,K),W)_K

and the Bernstein functor Π is defined by

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

References

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• Gregg J. Zuckerman, Construction of representations via derived functors, unpublished lecture series at the Institute for Advanced Study, 1978.