Archiv der Mathematik

, Volume 108, Issue 1, pp 99–112 | Cite as

On the Vergne conjecture

  • Peter HochsEmail author
  • Yanli Song


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).


Vergne conjecture Geometric quantisation Symplectic reduction 

Mathematics Subject Classification

Primary 53D50 Secondary 58J20 53D20 


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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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Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.University of AdelaideAdelaideAustralia
  2. 2.Dartmouth CollegeHanoverUSA

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