## Quantisation of presymplectic manifolds, $K$-theory and group representations

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- by Peter Hochs
- Proc. Amer. Math. Soc.
**143**(2015), 2675-2692 - DOI: https://doi.org/10.1090/S0002-9939-2015-12464-1
- Published electronically: January 21, 2015
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## 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.## References

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## Bibliographic Information

**Peter Hochs**- Affiliation: School of Mathematical Sciences, North Terrace Campus, The University of Adelaide, Adelaide SA 5005, Australia
- MR Author ID: 786204
- ORCID: 0000-0001-9232-2936
- Email: peter.hochs@adelaide.edu.au
- Received by editor(s): November 12, 2012
- Received by editor(s) in revised form: November 6, 2013, and January 24, 2014
- Published electronically: January 21, 2015
- Communicated by: Varghese Mathai
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**143**(2015), 2675-2692 - MSC (2010): Primary 53D50; Secondary 19K56, 22D25
- DOI: https://doi.org/10.1090/S0002-9939-2015-12464-1
- MathSciNet review: 3326046