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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
DOI: https://doi.org/10.1090/S0002-9939-2015-12464-1
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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  • [1] Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and $ K$-theory of group $ C^\ast $-algebras, $ C^\ast $-algebras: 1943-1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240-291. MR 1292018 (96c:46070), https://doi.org/10.1090/conm/167/1292018
  • [2] F. Bottacin, `A Marsden-Weinstein reduction theorem for presymplectic manifolds', http://www.dmi.unisa.it/people/bottacin/www/pubbl.htm.
  • [3] Ana Cannas da Silva, Yael Karshon, and Susan Tolman, Quantization of presymplectic manifolds and circle actions, Trans. Amer. Math. Soc. 352 (2000), no. 2, 525-552. MR 1714519 (2000j:53118), https://doi.org/10.1090/S0002-9947-99-02260-6
  • [4] Jérôme Chabert, Siegfried Echterhoff, and Ryszard Nest, The Connes-Kasparov conjecture for almost connected groups and for linear $ p$-adic groups, Publ. Math. Inst. Hautes Études Sci. 97 (2003), 239-278. MR 2010742 (2004j:19004), https://doi.org/10.1007/s10240-003-0014-2
  • [5] J. J. Duistermaat, The heat kernel Lefschetz fixed point formula for the spin-$ c$ Dirac operator, Progress in Nonlinear Differential Equations and their Applications, 18, Birkhäuser Boston Inc., Boston, MA, 1996. MR 1365745 (97d:58181)
  • [6] A. Echeverría-Enríquez, M. C. Muñoz-Lecanda, and N. Román-Roy, Reduction of presymplectic manifolds with symmetry, Rev. Math. Phys. 11 (1999), no. 10, 1209-1247. MR 1734712 (2001b:53106), https://doi.org/10.1142/S0129055X99000386
  • [7] Thomas Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, Providence, RI, 2000. Translated from the 1997 German original by Andreas Nestke. MR 1777332 (2001c:58017)
  • [8] Mark J. Gotay, James M. Nester, and George Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys. 19 (1978), no. 11, 2388-2399. MR 506712 (80e:58025), https://doi.org/10.1063/1.523597
  • [9] Michael Grossberg and Yael Karshon, Bott towers, complete integrability, and the extended character of representations, Duke Math. J. 76 (1994), no. 1, 23-58. MR 1301185 (96i:22030), https://doi.org/10.1215/S0012-7094-94-07602-3
  • [10] Michael D. Grossberg and Yael Karshon, Equivariant index and the moment map for completely integrable torus actions, Adv. Math. 133 (1998), no. 2, 185-223. MR 1604738 (2000f:53112), https://doi.org/10.1006/aima.1997.1686
  • [11] Nigel Higson and John Roe, Analytic $ K$-homology, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. Oxford Science Publications. MR 1817560 (2002c:58036)
  • [12] P. Hochs and N. P. Landsman, The Guillemin-Sternberg conjecture for noncompact groups and spaces, J. K-Theory 1 (2008), no. 3, 473-533. MR 2433278 (2010f:53160), https://doi.org/10.1017/is008001002jkt022
  • [13] Peter Hochs, Quantisation commutes with reduction at discrete series representations of semisimple groups, Adv. Math. 222 (2009), no. 3, 862-919. MR 2553372 (2011a:22016), https://doi.org/10.1016/j.aim.2009.05.011
  • [14] P. Hochs and V. Mathai, `Geometric quantization and families of inner products', arXiv:1309.6760.
  • [15] Yael Karshon and Susan Tolman, The moment map and line bundles over presymplectic toric manifolds, J. Differential Geom. 38 (1993), no. 3, 465-484. MR 1243782 (94j:58065)
  • [16] Anthony W. Knapp, Representation theory of semisimple groups, An overview based on examples. Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. MR 855239 (87j:22022)
  • [17] V. Lafforgue, Banach $ KK$-theory and the Baum-Connes conjecture (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 795-812. MR 1957086 (2003k:19006)
  • [18] N. P. Landsman, Functorial quantization and the Guillemin-Sternberg conjecture, Twenty years of Bialowieza: a mathematical anthology, World Sci. Monogr. Ser. Math., vol. 8, World Sci. Publ., Hackensack, NJ, 2005, pp. 23-45. MR 2181545 (2006h:58028)
  • [19] H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992 (91g:53001)
  • [20] Varghese Mathai and Weiping Zhang, Geometric quantization for proper actions, With an appendix by Ulrich Bunke. Adv. Math. 225 (2010), no. 3, 1224-1247. MR 2673729 (2011i:53149), https://doi.org/10.1016/j.aim.2010.03.023
  • [21] Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121-130. MR 0402819 (53 #6633)
  • [22] Xiaonan Ma and Weiping Zhang, Geometric quantization for proper moment maps, C. R. Math. Acad. Sci. Paris 347 (2009), no. 7-8, 389-394 (English, with English and French summaries). MR 2537236 (2010g:53176), https://doi.org/10.1016/j.crma.2009.02.003
  • [23] Xiaonan Ma and Weiping Zhang, Geometric quantization for proper moment maps: the Vergne conjecture, Acta Math. 212 (2014), no. 1, 11-57. MR 3179607, https://doi.org/10.1007/s11511-014-0108-3
  • [24] Eckhard Meinrenken, Symplectic surgery and the $ {\rm Spin}^c$-Dirac operator, Adv. Math. 134 (1998), no. 2, 240-277. MR 1617809 (99h:58179), https://doi.org/10.1006/aima.1997.1701
  • [25] Eckhard Meinrenken and Reyer Sjamaar, Singular reduction and quantization, Topology 38 (1999), no. 4, 699-762. MR 1679797 (2000f:53114), https://doi.org/10.1016/S0040-9383(98)00012-3
  • [26] Paul-Emile Paradan, Localization of the Riemann-Roch character, J. Funct. Anal. 187 (2001), no. 2, 442-509. MR 1875155 (2002m:53132), https://doi.org/10.1006/jfan.2001.3825
  • [27] Paul-Emile Paradan, Spin-quantization commutes with reduction, J. Symplectic Geom. 10 (2012), no. 3, 389-422. MR 2983435
  • [28] Paul-Émile Paradan, $ {\rm Spin}^c$-quantization and the $ K$-multiplicities of the discrete series, Ann. Sci. École Norm. Sup. (4) 36 (2003), no. 5, 805-845 (English, with English and French summaries). MR 2032988 (2004m:53159), https://doi.org/10.1016/j.ansens.2003.03.001
  • [29] Paul-Émile Paradan, Formal geometric quantization II, Pacific J. Math. 253 (2011), no. 1, 169-211. MR 2869441, https://doi.org/10.2140/pjm.2011.253.169
  • [30] P.-E. Paradan, `Quantization commutes with reduction in the noncompact setting: the case of the holomorphic discrete series', arXiv:1201.5451.
  • [31] M. G. Penington and R. J. Plymen, The Dirac operator and the principal series for complex semisimple Lie groups, J. Funct. Anal. 53 (1983), no. 3, 269-286. MR 724030 (85d:22016), https://doi.org/10.1016/0022-1236(83)90035-6
  • [32] Youliang Tian and Weiping Zhang, An analytic proof of the geometric quantization conjecture of Guillemin-Sternberg, Invent. Math. 132 (1998), no. 2, 229-259. MR 1621428 (2000d:53141), https://doi.org/10.1007/s002220050223
  • [33] Antony Wassermann, Une démonstration de la conjecture de Connes-Kasparov pour les groupes de Lie linéaires connexes réductifs, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 18, 559-562 (French, with English summary). MR 894996 (89a:22010)

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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

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