Localization and the quantization conjecture

Let (M, ω) be a compact symplectic manifold with a Hamiltonian action of a compact Lie group K. Suppose that 0 is a regular value of the moment map μ: M → Lie(K)∗, so that the Marsden-Weinstein reduction Mred = μ−(0)K is a symplectic orbifold. In our earlier paper (Quart. J. Math., 47, 1996) we proved a formula (the residue formula) for η0eω0[Mred] for any η0 ϵ H∗(Mred), where ω0 is the induced symplectic form on Mred. This formula is given in terms of the restrictions of classes in the equivariant cohomology H∗T(M) of M to the components of the fixed point set of a maximal torus T in M. s paper, we consider a line bundle L on Mfor which c1(L) = ω. If M is given a K-invariant complex structure compatible with ω we may apply the residue formula when η0 is the Todd class of Mred to obtain a formula for the Riemann-Roch number RR(Lred) of the induced line bundle Lred on Mred when K acts freely on μ−1(0). More generally when 0 is a regular value of μ, so that Mred is an orbifold and Lred is an orbifold bundle, Kawasakiʹs Riemann-Roch theorem for orbifolds can be applied, in combination with the residue formula. Using the holomorphic Lefschetz formula we similarly obtain a formula for the K-invariant Riemann-Roch number RRK(L) of L. We show that the formulae obtained for RR(Lred) and RRK(L) are almost identical and in many circumstances (including when K is a torus) are the same. Thus in these circumstances a special case of the residue formula is equivalent to the conjecture of Guillemin and Sternberg (Invent. Math. 67 (1982), 515–538) (proved in various degrees of generality by Guillemin and Sternberg themselves and others including Sjamaar, Guillemin, Vergne and Meinrenken) that RR(Lred) = RRK(L).