To the memory of Armand Borel GENERALIZED HARISH-CHANDRA MODULES WITH GENERIC MINIMAL k-TYPE

We make a first step towards a classification of simple generalized HarishChandra modules which are not Harish-Chandra modules or weight modules of finite type. For an arbitrary algebraic reductive pair of complex Lie algebras (g, k), 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 k-type. More precisely, we prove that any such simple (g, k)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 k contains a semisimple regular element of g, we prove that any simple (g, k)-module with generic minimal k-type is necessarily of finite type, and hence obtain a reconstruction theorem for a class of simple (g, k)-modules 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 k-type for any pair (g, k) with k ≃ sl(2).