Quantisation commutes with reduction for cocompact Hamiltonian group actions

Abstract

The Guillemin--Sternberg conjecture states that `quantisation commutes with reduction' for Hamiltonian actions by compact Lie groups on compact symplectic manifolds. In the most general form of this conjecture, the quantisation of such an action is defined as the equivariant index of a Dolbeault--Dirac or Spin-c-Dirac operator. This form of the conjecture has been proved in various ways, and in various degrees of generality, by several authors in the 1990s. Recently, Landsman proposed a generalisation of the Guillemin--Sternberg conjecture to actions by possibly noncompact Lie groups on possibly noncompact symplectic manifolds, as long as the orbit space of the action is compact. In this generalisation, the representation ring of the group in question is replaced by the K-theory of its C*-algebra, and the equivariant index is replaced by the analytic assembly map that is used in the Baum--Connes conjecture in noncommutative geometry. In this thesis, we prove Landsman's generalisation in two cases. In the first case, we consider reduction at the trivial representation, for groups with a cocompact, discrete normal subgroup. In the second case, we consider reduction at discrete series representations of semisimple groups.