Multiplicity Formulae for Discrete Series.

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