Multiplicity formulae for discrete series

Abstract

In a series of papers [20, 211 and [22], Schmid obtained several important results on the discrete series for semisimple Lie groups. The purpose of this paper is to prove Schmid's results by somewhat different methods and to relax as much as possible the restriction imposed on the regularity of the parameters of discrete classes. Though the basic line is similar to Schmid's, the main difference is that our methods do not rely upon complex analysis on the non-compact flag manifold G/T as developed in [20] and [21]. Rather, we just rely on some elementary differential calculus on the symmetric space G/K. This difference gives rise to less restrictive assumptions since the results on the vanishing of L2-cohomologies in the symmetric space situation seem to be sharper, so far. Our work also leads to some interesting results concerning the multiplicity formula of discrete classes in L 2 (/' \ G). In our development, we shall give an alternative proof of the key fact (Theorem 1, w 4) which is obtained in [20] using the complex analysis on G/T. Our proof, given in w 5, will be carried out directly in the symmetric space situation, and some standard theory of sheaf cohomology centering the Borel-WeilBott theorem on the compact flag manifold KIT will be used as in [20]. In this proof, our methods seem to be quite elementary. We now give a more precise description of the contents of the paper. Let G be a non-compact real semisimple Lie group with discrete series g2 4= qS. Assume, for simplicity throughout the paper, that G is a connected real form of a simply connected complex semisimple Lie group G e. Harish-Chandra 1-7] showed that there exists a compact Cartan subgroup T of G and that, if one denotes by T' the set of regular characters of T, there exists a distinguished surjection co: T'---, g2. We fix T and a maximal compact subgroup K containing T once and for all. Let ge, t e denote the complexifications of the Lie algebras g, t of G, T respectively. In considering the discrete class co (A) for a given regular character A e T', we always choose a positive root system P for the pair (go, t~) such that A is regular dominant with respect to 19, i.e., P = {~; (A, c0>0 }. Here as