[Q,R]=0 and Kostant partition functions

TL;DR

This paper offers a new proof that the dimension of invariant functions on certain symplectic manifolds varies polynomially with degree, using Atiyah-Bott fixed point formula and partition functions.

Contribution

It introduces a novel proof of polynomiality in symplectic geometry using fixed point formulas and localization techniques, extending previous positivity-based results.

Findings

Proves polynomial dependence of invariant function dimensions on degree k

Utilizes Atiyah-Bott fixed point formula in the proof

Develops a localization method for positivity in symplectic geometry

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. 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.