Quantisation commutes with reduction at discrete series representations of semisimple groups

Abstract

Using the analytic assembly map that appears in the Baum-Connes conjecture in noncommutative geometry, we generalise the Guillemin-Sternberg conjecture that 'quantisation commutes with reduction' to (discrete series representations of) semisimple groups G with maximal compact subgroups K acting cocompactly on symplectic manifolds. We prove this generalised statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements gse*, the set of elements of g* with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that M = G ×K N, for a compact Hamiltonian K-manifold N. The proof comes down to a reduction to the compact case. This reduction is based on a 'quantisation commutes with induction'-principle, and involves a notion of induction of Hamiltonian group actions. This principle, in turn, is based on a version of the naturality of the assembly map for the inclusion K {right arrow, hooked} G. © 2009 Elsevier Inc. All rights reserved.