GENERALIZED HARISH-CHANDRA MODULES

Let g be a complex reductive Lie algebra and h be a Cartan subalgebra of g. If t is a subalgebra of g, we call a g-module M a strict (g, t)-module if t coincides with the subalgebra of all elements of g which act locally finitely on M. For an intermediate t, i.e., such that h subset of t subset of g, we construct irreducible strict (g, t)-modules. The method of construction is based on the D-module localization theorem of Beilinson and Bernstein. The existence of irreducible strict (g, t)-modules has been known previously only for very special subalgebras t, for instance when t is the (reductive) subalgebra of fixed points of an involution of g. In this latter case strict irreducible (g, t)-modules are Harish-Chandra modules. We also give separate necessary and sufficient conditions on t for the existence of an irreducible strict (g, t)-module of finite type, i.e., an irreducible strict (g, t)-module with finite t-multiplicities. In particular, under the assumptions that the intermediate subalgebra t is reductive and g has no simple components of types B-n for n > 2 or F-4, we prove a simple explicit criterion on t for the existence of an irreducible strict (g, t)-module of finite type. It implies that, if g is simple of type A or C, for every reductive intermediate t there is an irreducible strict (g, t)-module of finite type.