Quantisation of presymplectic manifolds, K-theory and group representations

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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×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.