Quantisation commutes with reduction at discrete series representations of semisimple groups

TL;DR

This paper extends the 'quantisation commutes with reduction' principle to discrete series representations of semisimple groups using noncommutative geometry tools, proving the conjecture under specific ellipticity conditions.

Contribution

It generalizes the Guillemin-Sternberg conjecture to semisimple groups with a new proof based on the Baum-Connes assembly map and induction principles.

Findings

Proves quantisation commutes with reduction for certain semisimple group actions.

Introduces a reduction technique to the compact case via induction.

Establishes naturality of the assembly map for group inclusions.

Abstract

Using the analytic assembly map that appears in the Baum-Connes conjecture in noncommutative geometry, we generalise the $\Spin^c$-version of 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 statement in cases where the image of the momentum map in question lies in the set of strongly elliptic elements, the set of elements of $\g^*$ with compact stabilisers. This assumption on the image of the momentum map is equivalent to the assumption that $M = G \times_K N$, 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 of $K$ into $G$.