Generalized Harish-Chandra Modules with Generic Minimal t-Type

Abstract

We make a first step towards a classification of simple generalized Harish-Chandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras(g,t), we construct, via cohomological induction, the fundamental series F ^. (p,E) of generalized Harish-Chandra modules. We then use F^. (p,E) to characterize any simple generalized Harish-Chandra module with generic minimal t-type. More precisely, we prove that any such simple(g,t)-module of finite type arises as the unique simple submodule of an appropriate fundamental series module F^s (p,E) in the middle dimension s. Under the stronger assumption that t contains a semisimple regular element of g, we prove that any simple(g,t)-module with generic minimal t-type is necessarily of finite type, and hence obtain a reconstruction theorem for a class of simple(g,t)-module which can a priori have infinite type. We also obtain generic general versions of some classical theorems of Harish-Chandra, such as the Harish-Chandra admissibility theorem. The paper is concluded by examples, in particular we compute the genericity condition on a ttype for any pair (g,t)with t about equal to sl(2).