The equivariant Riemann–Roch theorem and the graded Todd class

Abstract

Let G be a torus with Lie algebra g and let M be a G-Hamiltonian manifold with Kostant line bundle L and proper moment map. Let Λ⊂g⁎ be the weight lattice of G. We consider a parameter k≥1 and the multiplicity m(λ,k) of the quantized representation RRG(M,Lk). Define 〈Θ(k),f〉=∑λ∈Λm(λ,k)f(λ/k) for f a test function on g⁎. We prove that the distribution Θ(k) has an asymptotic development 〈Θ(k),f〉∼kdim⁡M/2∑n=0∞k−n〈DHn,f〉 where the distributions DHn are the twisted Duistermaat–Heckman distributions associated with the graded equivariant Todd class of M. When M is compact, and f polynomial, the asymptotic series is finite and exact.