Harish-Chandra systems on a reductive Lie algebra and the Zuckerman functor

Let g be a complex reductive Lie algebra with adjoint group G, Cartan subalgebra t and Weyl group W. Harish-Chandra defined a homomorphism {dollar}delta:{lcub}cal D{rcub}({lcub}rmbf g{rcub})sp{lcub}G{rcub}to{lcub}cal D{rcub}({lcub}bf t{rcub})sp{lcub}W{rcub}{dollar} of algebras of invariant polynomial differential operators. In a recent joint paper, Wallach and I have given an algebraic construction of {dollar}delta{dollar} analogous to that of the Harish-Chandra isomorphism from the center of the universal enveloping algebra of g to the Weyl group invariants in the symmetric algebra of t. The proof that our construction does indeed yield {dollar}delta{dollar} involves the analysis of a certain {dollar}{lcub}cal D{rcub}{dollar}(g)-module {dollar}{lcub}cal M{rcub}.{dollar} For the algebra {dollar}{lcub}cal D{rcub}({lcub}rmbf g{rcub})sp{lcub}G{rcub}{dollar} of invariant differential operators this module is analogous to a Verma module for the Lie algebra g.; In this thesis we introduce a family of {dollar}{lcub}cal D{rcub}{dollar}(g)-modules, {dollar}{lcub}cal M{rcub}sb{lcub}lambda{rcub},{dollar} parametrized by the elements {dollar}lambdain{lcub}bf t{rcub}sp{lcub}*{rcub}.{dollar} The module corresponding to {dollar}lambda=0{dollar} is the module {dollar}{lcub}cal M{rcub}{dollar} from above. If {dollar}lambdanot=0,{dollar} then {dollar}{lcub}cal M{rcub}sb{lcub}lambda{rcub}{dollar} as a module for {dollar}{lcub}cal D{rcub}({lcub}bf g{rcub})sp{lcub}G{rcub}{dollar} is analogous to a Whittaker module for g. Let {dollar}tau:{lcub}bf g{rcub}to{lcub}cal D{rcub}({lcub}bf g{rcub}){dollar} be the embedding of g inside {dollar}{lcub}cal D{rcub}{dollar}(g) as adjoint vector fields. Our main result consists of relating {dollar}{lcub}cal M{rcub}sb{lcub}lambda{rcub}{dollar} to the invariant holonomic system{dollar}{dollar}{lcub}cal N{rcub}sb{lcub}lambda{rcub}={lcub}cal D{rcub}({lcub}bf g{rcub})/({lcub}cal D{rcub}({lcub}bf g{rcub})tau({lcub}bf g{rcub})+sumlimitssb{lcub}Pin S({lcub}bf g{rcub})sp{lcub}G{rcub}{rcub}{lcub}cal D{rcub}({lcub}bf g{rcub})(P-P(lambda)).{dollar}{dollar}We show that {dollar}{lcub}cal N{rcub}sb{lcub}lambda{rcub}{dollar} is obtained from {dollar}{lcub}cal M{rcub}sb{lcub}lambda{rcub}{dollar} by applying an appropriate Zuckerman functor. Our construction unifies a geometric approach to invariant holonomic systems due to Hotta and Kashiwara, and a recent algebraic approach due to Levasseur and Stafford. The connection with the {dollar}{lcub}cal D{rcub}{dollar}-module constructions of Hotta and Kashiwara is made possible through an idea of Evens and uses the geometric interpretation of the Zuckerman functor due to Bernstein.; As an application of our techniques, we give a formula for the graded character of the system {dollar}{lcub}cal N{rcub}sb0.{dollar} This formula defines for each irreducible character of G a Laurent polynomial with non-negative integer coefficients. There are connections with Lusztig's q-analogs of weight multiplicity.