Quantisation of presymplectic manifolds, 𝐾-theory and group representations

Source abstract

Let G G be a semisimple Lie group with finite component group, and let K > G K>G be a maximal compact subgroup. We obtain a quantisation commutes with reduction result for actions by G G on manifolds of the form M = G × K N M = G\times _K N , where N N is a compact prequantisable Hamiltonian K K -manifold. The symplectic form on N N induces a closed two-form on M 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 K -theory of the C ∗ C^* -algebra of a semisimple group are realised as quantisations of fibre bundles over suitable coadjoint orbits.