Quantisation commutes with reduction at discrete series representations of semisimple groups

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 $G$.Comment: 60 pages, substantial error correcte

Quantisation of presymplectic manifolds, K-theory and group representations

Let $G$ be a semisimple Lie group with finite component group, and let $K<G$ be a maximal compact subgroup. We obtain a quantisation commutes with reduction result for actions by $G$ on manifolds of the form $M = G\times_K N$, where $N$ is a compact prequantisable Hamiltonian $K$-manifold. The symplectic form on $N$ induces a closed two-form on $M$, which may be degenerate. We therefore work with presymplectic manifolds, where we take a presymplectic form to be a closed two-form. For complex semisimple groups and semisimple groups with discrete series, the main result reduces to results with a more direct representation theoretic interpretation. The result for the discrete series is a generalised version of an earlier result by the author. In addition, the generators of the $K$-theory of the $C^*$-algebra of a semisimple group are realised as quantisations of fibre bundles over suitable coadjoint orbits

Geometric quantization and families of inner products

We formulate a quantization commutes with reduction principle in the setting where the Lie group $G$, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and an equivariant family of inner products on the Lie algebra $\mathfrak{g}$ of $G$, is $G$-cocompact. The central result establishes an asymptotic version of this quantization commutes with reduction principle. Using an equivariant family of inner products on $\mathfrak{g}$ instead of a single one makes it possible to handle both noncompact groups and manifolds, by extending Tian and Zhang's Witten deformation approach to the noncompact case.Comment: 72 pages. Minor corrections after comments from a refere