Generalized Harish-Chandra Modules with Generic Minimal <b>t</b>-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,k), we construct, via cohomological induction, the fundamental series F · (p ,E ) of generalized Harish-Chandra modules. We then use F · (p ,E )t o char- acterize 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 ac lass of simple (g,k)-modules which can a priori have infinite type. We also obtain generic gen- eral 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 )w ithks� (2). Introduction. The goal of the present paper is to 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. This work is part of the program of study of generalized Harish-Chandra modules laid out in (PZ). Let g be a semisimple Lie algebra. A simple generalized Harish-Chandra module is by definition a simple g-module with locally finite action of a reductive in g subalgebra k ⊂ g and with finite k-multiplicities. In the classical case of Harish-Chandra modules, the pair (g,k )i s in addition assumed to be symmetric. In a recent joint paper with V. Serganova (PSZ), we have constructed new families of generalized Harish-Chandra modules; however, no general classification is known beyond the case when the pair (g,k) is symmetric and the case when k is a Cartan subalgebra of g. The first case is settled in the well- known work of R. Langlands (L2), A. Knapp and the second named author (KZ), D. Vogan and the second named author (V2), (Z), A. Beilinson - J. Bernstein (BB) and I. Mirkovic (Mi); the second case is settled in a more recent breakthrough by O. Mathieu (M). Some low rank cases of certain special non-symmetric pairs (g,k) (where k is not a Cartan subalgebra) have been settled by G. Savin (Sa). In this paper, we consider simple generalized Harish-Chandra modules which have a generic minimal k-type for some arbitrary fixed reductive pair (g,k) (the precise definitions see in Section 1 below). One of our main results is the construction of as eries of (g,k)-modules, which we call the fundamental series (it generalizes the fundamental series of Harish-Chandra modules), and in addition the theorem that any simple generalized Harish-Chandra module with generic minimal k-type is a submodule of the fundamental series. We refer to the latter result as the first reconstruction theorem for generalized Harish-Chandra modules. This theorem is based on new