Zuckerman functor

• 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.

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

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