Quantisation of presymplectic manifolds, K-theory and group representations

TL;DR

This paper establishes a quantisation with reduction principle for presymplectic manifolds acted upon by semisimple Lie groups, linking geometric quantisation to group representations and K-theory.

Contribution

It extends quantisation and reduction results to presymplectic manifolds with group actions, connecting geometric quantisation to representation theory and K-theory.

Findings

Quantisation commutes with reduction for certain group actions on presymplectic manifolds.

Realisation of K-theory generators as quantisations of fibre bundles over coadjoint orbits.

Reduction results specialize to known cases with direct representation theoretic interpretations.

Abstract

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.