  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.
