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Using the analytic assembly map that appears in the Baum-Connes conjecture innoncommutative geometry, we generalise the $\Spin^c$-version of theGuillemin-Sternberg conjecture that `quantisation commutes with reduction' to(discrete series representations of) semisimple groups $G$ with maximal compactsubgroups $K$ acting cocompactly on symplectic manifolds. We prove thisstatement in cases where the image of the momentum map in question lies in theset of strongly elliptic elements, the set of elements of $\g^*$ with compactstabilisers. This assumption on the image of the momentum map is equivalent tothe 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 isbased on a `quantisation commutes with induction'-principle, and involves anotion of induction of Hamiltonian group actions. This principle, in turn, isbased on a version of the naturality of the assembly map for the inclusion of$K$ into $G$.

Quantisation commutes with reduction at discrete series representations of semisimple groups