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Holomorphic Quantization and Unitary Representations of the Teichmüller Group

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Lie Theory and Geometry

Part of the book series: Progress in Mathematics ((PM,volume 123))

Abstract

One of the purposes of geometric quantization (in the sense of Kostant) is to associate in some intrinsic way a finite-dimensional Hilbert space E to a compact symplectic manifold (M, ω) such that \( 2\pi \sqrt {{ - 1}} .\omega \) is the curvature of a connection ∇ on some line bundle L. It is usually assumed that M is simply-connected, since then the pair (L, ∇) is unique up to isomorphism, as was shown in [Ko]. In case there is a finite-dimensional Lie group G of symplectomorphisms present, a central extension \( \mathop{G}\limits^{ \sim } \) of G was constructed by Kostant [Ko], and this central extension should act on E. Thus Kostant was able to recover the Borel-Weil-Bott theorem from the action of a compact Lie group G on a quantizable coadjoint orbit.

Each author was supported in part by a grant from the NSF.

Dedicated to Bert Kostant with admiration

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Author information

Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Jean-Luc Brylinski

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Dennis McLaughlin

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  1. Jean-Luc Brylinski
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  2. Dennis McLaughlin
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Editors and Affiliations

  1. Department of Mathematics, Penn State University, 16802, University Park, PA, USA

    Jean-Luc Brylinski  & Ranee Brylinski  & 

  2. Department of Mathematics, MIT, 02139, Cambridge, MA, USA

    Victor Guillemin  & Victor Kac  & 

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Brylinski, JL., McLaughlin, D. (1994). Holomorphic Quantization and Unitary Representations of the Teichmüller Group. In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics, vol 123. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0261-5_2

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  • DOI: https://doi.org/10.1007/978-1-4612-0261-5_2

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6685-3

  • Online ISBN: 978-1-4612-0261-5

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