The equivariant Riemann-Roch theorem and the graded Todd class

TL;DR

This paper establishes an asymptotic expansion relating the multiplicities of quantized representations on G-Hamiltonian manifolds to twisted Duistermaat-Heckman distributions, generalizing the Riemann-Roch theorem in an equivariant setting.

Contribution

It introduces a new asymptotic formula connecting representation multiplicities with the graded Todd class for G-Hamiltonian manifolds, extending classical Riemann-Roch results.

Findings

Asymptotic expansion for multiplicities in terms of twisted Duistermaat-Heckman distributions.

Exact finite series for polynomial test functions on compact manifolds.

Generalization of the equivariant Riemann-Roch theorem to non-compact settings.

Abstract

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 $P/k$ 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.