Œ[$Q, R$] = 0 and Kostant partition functions

Abstract

On a polarized compact symplectic manifold endowed with an action of a compact Lie group, in analogy with geometric invariant theory, one can define the space of invariant functions of degree $k$. A central statement in symplectic geometry, the quantization commutes with reduction hypothesis, is equivalent to saying that the dimension of these invariant functions depends polynomially on $k$. This statement was proved by Meinrenken and Sjamaar under positivity conditions. In this paper, we give a new proof of this polynomiality property based on a study of the Atiyah–Bott fixed point formula from the point of view of the theory of partition functions, and a technique for localizing positivity.