  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 $G$.
