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 . 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 . 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.
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András Szenes, Michèle Vergne,  = 0 and Kostant partition functions. Enseign. Math. 63 (2017), no. 3/4, pp. 471–516DOI 10.4171/LEM/63-3/4-8