On the Vergne conjecture

Abstract

Consider a Hamiltonian action by a compact Lie group on a possibly non-compact symplectic manifold. We give a short proof of a geometric formula for the decomposition into irreducible representations of the equivariant index of a \({{\mathrm{{{\mathrm{Spin}}}^c}}}\)-Dirac operator in this context. This formula was conjectured by Vergne in (Eur Math Soc Zürich I:635–664, 2007) and proved by Ma and Zhang in (Acta Math 212:11–57, 2014).

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Acknowledgements

The authors would like to thank Maxim Braverman, Nigel Higson, Paul-Émile Paradan, Eckhard Meinrenken, and Michèle Vergne for many useful discussions. Special thanks go to Xiaonan Ma and Weiping Zhang for proposing this topic and kind help. Peter Hochs was supported by the European Union, through Marie Curie fellowship PIOF-GA-2011-299300.

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Hochs, P., Song, Y. On the Vergne conjecture. Arch. Math. 108, 99–112 (2017). https://doi.org/10.1007/s00013-016-0997-9

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Mathematics Subject Classification

  • Primary 53D50
  • Secondary 58J20
  • 53D20

Keywords

  • Vergne conjecture
  • Geometric quantisation
  • Symplectic reduction