On an Infinitesimal Characterization of the Discrete Series

Abstract

The aim of this paper is to obtain an infinitesimal characterization of the discrete series of representations of a semisimple Lie group. Let G be a semisimple Lie group, connected and having a finite center. For simplicity we assume that G is contained as a real form of a complex simply connected semisimple Lie group G,. Let K c G be a maximal compact subgroup of G. We assume that rk (G) = rk (K), rk denoting the rank, and fix a maximal torus B c K. Let g be the Lie algebra of G, and f, b, the subalgebras defined by K, B respectively. g, is the complexification of g (g c g,), and (5 is the universal enveloping algebra of ge, while A is the subalgebra of (3 generated by (1, f). A is the set of roots of (g, b), and A, those of (f, b); Pk is a fixed positive system in Ak. L is the lattice of all integral linear functions on bc, i.e., X e bc* such that 2 / e Z for all a e A. L' is the subset of all X e L such that O for all a e A. L, is the set of all X e VC such that 2 / e Z for all f e Ak; C, is the set of all X e b* such that is real and ?0 for all eak. We write L+ = L, n Ca. For any te e Lt, z-t is the unique irreducible St-module with highest weight 4ei (relative to Pk). If se e L n Lt (and only for such ts), z,> gives rise to a K-module, also denoted by zry g = f + 4p is the Cartan decomposition of g. Finally, 3 denotes the center of $ and ?A, the centralizer of K in 6. The elements of L' n Cf parameterize the equivalence classes of the discrete series of representations of G (cf. [6]; see also Section 6). For any A e L' n Cf let co(A) denote the corresponding equivalence class. Fix A e L' n C, and let P be the positive system of roots of (g, b) such that > 0 for all a e P. Define ak, a, by (19) of Section 6. Then A 8k + an e L n Lt and one knows that at least when all the numbers (a e P) are sufficiently large, the discrete class co(AX) contains ZA-3k+an with multiplicity 1 ([8], [9]). By Schur's lemma the elements of D act as scalars on the corresponding isotypical subspace, giving rise to a homomorphism XA of Z into C. A well-known theorem of Harish-Chandra [4] now implies that