On the Vergne conjecture

Consider a Hamiltonian action by a compact Lie group on a possibly noncompact symplectic manifold. We give a short proof of a geometric formula for decomposition into irreducible representations of the equivariant index of a Spin${}^c$-Dirac operator in this context. This formula was conjectured by Mich`ele Vergne in 2006 and proved by Ma and Zhang in 2014.