JournalslemVol. 63, No. 3/4pp. 471–516

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

  • András Szenes

    Université de Genève, Switzerland
  • Michèle Vergne

    Université Paris-Diderot Paris 7, Paris, France
Œ[$Q, R$] = 0 and Kostant partition functions cover
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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 kk. 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 kk. 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, Œ[Q,RQ, R] = 0 and Kostant partition functions. Enseign. Math. 63 (2017), no. 3, pp. 471–516

DOI 10.4171/LEM/63-3/4-8