On localization and Riemann-Roch numbers for symplectic quotients

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