  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.
The proof is 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.
