We establish an analytic interpretation for the index of certain transversally elliptic symbols on non-compact manifolds. By using this interpretation, we establish a geometric quantization formula for a Hamiltonian action of a compact Lie group acting on a non-compact symplectic manifold with proper moment map. In particular, we present a solution to a conjecture of Michèle Vergne in her ICM 2006 plenary lecture.
"Geometric quantization for proper moment maps: the Vergne conjecture." Acta Math. 212 (1) 11 - 57, 2014. https://doi.org/10.1007/s11511-014-0108-3