Localization and the quantization conjecture

Abstract

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 M red = μ −(0) K is a symplectic orbifold. In our earlier paper ( Quart. J. Math., 47, 1996) we proved a formula (the residue formula) for η 0 e ω0 [ M red ] for any η 0 ϵ H ∗(M red) , where ω 0 is the induced symplectic form on M red . 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. In this paper, we consider a line bundle L on Mfor which c 1( L) = ω . If M is given a K-invariant complex structure compatible with ω we may apply the residue formula when η 0 is the Todd class of M red to obtain a formula for the Riemann-Roch number RR( L red) of the induced line bundle L red on M red when K acts freely on μ −1(0). More generally when 0 is a regular value of μ, so that M red is an orbifold and L red is an orbifold bundle, Kawasaki's 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 RR K( L) of L . We show that the formulae obtained for RR( L red) and RR K( 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( L red) = RR K( L) .