On localization and Riemann-Roch numbers for symplectic quotients

Abstract

Suppose $(M,\omega)$ is a compact symplectic manifold acted on by a compact Lie group $K$ in a Hamiltonian fashion, with moment map $\mu: M \to \Lie(K)^*$ and Marsden-Weinstein reduction $M_{red} = \mu^{-1}(0)/K$. In this paper, we assume that $M$ has a $K$-invariant K\"ahler structure. In an earlier paper, we proved a formula (the residue formula) for $\eta_0 e^{\omega_0}[M_{red}]$ for any $\eta_0 \in H^*(M_{red})$, where $\omega_0$ is the induced symplectic form on $M_{red}$. Here we apply the residue formula in the special case $\eta_0 = Td(M_{red})$; when $K$ acts freely on $\mu^{-1}(0)$ this yields a formula for the Riemann-Roch number $RR (L_{red})$ of a holomorphic line bundle $L_{red}$ on $M_{red}$ that descends from a holomorphic line bundle $L$ on $M$ for which $c_1(L) = \omega$. Using the holomorphic Lefschetz formula we similarly obtain a formula for the $K$-invariant Riemann-Roch number $RR^K(L) $ of $L$. In the case when the maximal torus $T$ of $K$ has dimension one (except in a few special circumstances), we show the two formulas are the same. Thus in this special case the residue formula is equivalent to the result of Guillemin and Sternberg that $RR(L_{red}) = RR^K(L)$. (The residue formula was proved under the assumption that 0 is a regular value of $\mu$, and was given in terms of the restrictions of classes in the equivariant cohomology $H^*_T(M) $ of $M$ to the