The equivariant Riemann-Roch theorem and the graded Todd class

Let G be a torus and M a G-Hamiltonian manifold with Kostant line bundle L and proper moment map. Let P be the weight lattice of G. We consider a parameter k and the multiplicity $m(\lambda ,k)$ of the quantized representation associated to M and the k-th power of L . We prove that the weighted sum $\sum m(\lambda,k) f(\lambda/k)$ of the value of a test function f on points of the lattice $P\mathrm{/}k$ has an asymptotic development in terms of the twisted Duistermaat-Heckman distributions associated to the graded Todd class of M. When M is compact, and f polynomial, the asymptotic series is finite and exact.