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 Zurich I:635–664, 2007) and proved by Ma and Zhang in (Acta Math 212:11–57, 2014).