On the Vergne conjecture

TL;DR

This paper provides a concise proof of a geometric formula for decomposing the equivariant index of a Spin$^c$-Dirac operator under a Hamiltonian group action on noncompact symplectic manifolds, confirming a conjecture by Vergne.

Contribution

The paper offers a simplified proof of Vergne's 2006 conjectured formula for the equivariant index decomposition, building on prior work by Ma and Zhang.

Findings

Confirmed Vergne's conjecture with a new proof

Simplified the understanding of equivariant index decomposition

Extended the formula to possibly noncompact symplectic manifolds

Abstract

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.