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


Author: Peter Hochs
Journal: Proc. Amer. Math. Soc. 143 (2015), 2675-2692
MSC (2010): Primary 53D50; Secondary 19K56, 22D25
Published electronically: January 21, 2015
MathSciNet review: 3326046
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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.


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Additional Information

Peter Hochs
Affiliation: School of Mathematical Sciences, North Terrace Campus, The University of Adelaide, Adelaide SA 5005, Australia
Email: peter.hochs@adelaide.edu.au

DOI: https://doi.org/10.1090/S0002-9939-2015-12464-1
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
Article copyright: © Copyright 2015 American Mathematical Society