A Riemann-Roch formula for singular reductions by circle actions

We compute a Riemann-Roch formula for the invariant Riemann-Roch number of a quantizable Hamiltonian $S^1$-manifold $(M,\omega ,\mathcal{J})$ in terms of the geometry of its symplectic quotient, allowing $0$ to be a singular value of the moment map $\mathcal{J}:M\to \mathbb{R}$. The formula involves a new explicit local invariant of the singularities. Our approach relies on a complete singular stationary phase expansion of the associated Witten integral.