lnventiones math. 30, 47- 144 (1975) 9 by Springer-Verlag I975 On the Characters of the Discrete Series The Hermitian Symmetric Case Wilfried Schmid* (New York) Table of Contents w1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 47 w Cartan Subalgebras and Subgroups . . . . . . . . . . . . . 54 w The "Holomorphic Discrete Series" . . . . . . . . . . . . . 69 w The Characters of the Discrete Series . . . . . . . . . . . . 73 w Some Inductive Arguments . . . . . . . . . . . . . . . . . 80 w The Proofs of the Main Theorems . . . . . . . . . . . . . . 103 w Blattner's Conjecture . . . . . . . . . . . . . . . . . . . 122 w Explicit Realization of the Discrete Series . . . . . . . . . . 130 w Some Postscripts . . . . . . . . . . . . . . . . . . . . . 140 References . . . . . . . . . . . . . . . . . . . . . . . . 142 w 1. Introduction An irreducible, unitary representation of a locally compact, unimodular group is said to be square-integrable if it can be realized on an invariant subspace for the left regular representation. The discrete series is the set of equivalence classes of such representations. In Harish-Chandra's work on the Plancherel formula for semisimple Lie groups, the discrete series plays the central role. Thus, among the representations of a semisimple Lie group, the discrete series repre- sentations are of particular interest. According to Harish-Chandra's criterion [13], a connected semisimple Lie group G has a non-empty discrete series exactly when it contains a compact Cartan subgroup. If G does contain a compact Cartan subgroup H, Harish- Chandra parameterizes the discrete series by, roughly speaking, the dual group/q of H, modulo the action of the normalizer of H in G. To be more precise, 1 denote the Lie algebras of G, H by 9, D, and their complexifications by 94, If. Via ex- ponentiation, /t becomes isomorphic to a lattice Acil)*(D*=dual space of D); the lattice A contains the root system 4~ of (94, t)r For simplicity, assume that G has a complexification GC, which is simply connected. Then, according to Harish- Chandra's fundamental results on the discrete series [13], for every nonsingular 1 2e A, there exists a unique tempered 2 invariant eigendistribution Oa, such that (1.1) Oalit = ( - 1)q 1-1 (e~/2 ; I 1 ~ ~/,,(c,,.a.) > o -- e-•/2) * Supported in part by the Sonderforschungsbereich 40 (Reine Mathematik) at the University of Bonn, and by NSF Grant GP 32843. t i.e. (2, cr for all c~e~. 2 A distribution is called tempered if it extends continuously to the space of rapidly decreasing func- tions [ 13].