A COMPARISON THEOREM FOR n - HOMOLOGY

Introduction. The purpose of this paper is to compare homological properties of an analytic representation of a semisimple Lie group and of its Harish-Chandra module. Throughout the paper G0 denotes a connected semisimple Lie group with finite center. Let π be an admissible representation of G0 on a complete, locally convex Hausdorff topological vector space Mπ . Vectors in Mπ transforming finitely under the action of a maximal compact subgroup K0 of G0 are analytic and they form a subspace M invariant under the action of the Lie algebra g0 of G0 – this subspace is, by definition, the HarishChandra module of π. The advantages of working with the underlying Harish-Chandra module, rather than with the global representation itself, are multifold. Although not a G0-invariant subspace of Mπ , M retains the essential features of the representation. For example, it fully determines the (distributional) character of π. On the other hand, M is a much smaller object than Mπ: it is stripped of the often cumbersome functional analytic superstructure of the latter, and enjoys finiteness properties which make it amenable to algebraic, and, in particular, homological methods. In the study ofM a special role is played by the homology groupsHp(n,M), with respect to maximal nilpotent subalgebras n of the complexification g of g0. They carry information not only about the module itself, but also about the global representation. Two examples, somewhat interconnected, may serve as evidence: (1) For a generic n, H0(n,M) determines the growth of matrix coefficients of Mπ ([4], [6], [12], [15]). (2) The value of the character of Mπ at a regular point g ∈ G0 is equal to the “Euler characteristic” of the n-homology of M , where the choice of n depends on g ([11], see also [20]). These examples suggest that the groups Hp(n,M) are global invariants, and a natural question which arises is whether for a suitably chosen Mπ the map