A proof of Blattner's conjecture

Abstract

Let G be a connected, semisimple Lie group. In Harish-Chandra's work on the Plancherel formula, the discrete series of irreducible unitary representations [5] plays a crucial role. Roughly speaking, besides the discrete series itself, the representations which enter the Plancherel decomposition of L2(G) can all be built up from discrete series representations of subgroups of G. Among the various irreducible unitary representations, the discrete series representations are therefore of particular importance. The main technique in the representation theory of compact Lie groups is to investigate the restriction of any given representation to a maximal torus. Similarl~r to understand the structure of an irreducible representation of a noncompact semisimple Lie group, one may try to determine its restriction to a maximal compact subgroup. Blattner's conjecture predicts how the discrete series representations should break up under the action of a maximal compact subgroup. It formally resembles the formula for the multiplicity of a weight for finite-dimensional representations. A companion statement to Blattner's conjecture, analogous to the theorem of the highest weight, characterizes discrete series representations, up to infinitesimal equivalence, in terms of their restriction to a maximal compact subgroup [13]. Once and for all, we make the assumption that G contains a compact Cartan subgroup. Exactly in this situation G has a non-empty discrete series [5]. Let K be a maximal compact subgroup, with HcK. The Lie algebras of G, K, H will be written as go, ~0, bo, and their complexifications as g, [, b. A non-zero root of (g, b) is called compact or noncompact, depending on whether or not its root space lies in L We denote the sets of compact and noncompact roots by, respectively, q~ and 4"; ~ = ~c w 4" is thus the setof all non-zero roots of (g, b). The Killing form of g induces an inner product ( , ) on ib*, the space of linear functions on b which assume imaginary values on b0An element ~teib* is said to be singular if it is perpendicular to at least one ct~q~, and nonsingular otherwise. The differentials of the characters of H form a lattice A cib*, which contains ~b. Since