GENERALIZED HARISH-CHANDRA MODULES WITH GENERIC MINIMAL k-TYPE

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) 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). 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 HarishChandra 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) is 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 wellknown 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 a series 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 ∗ Received August 12, 2004; accepted for publication September 15, 2004. † International University Bremen, Campus Ring I, D-28759 Bremen, Germany ([email protected]). ‡ Department of Mathemaics, Yale University, 10 Hillhouse Ave., P.O. Box 208283, New Haven, CT 06520-8283, USA ([email protected]).