Integrals of equivariant forms over noncompact symplectic manifolds

Abstract

This article is a result of the AIM workshop on Moment Maps and Surjectivity in Vari- ous Geometries (August 9 - 13, 2004) organized by T. Holm, E. Lerman and S. Tolman. At that workshop I was introduced to the work of T. Hausel and N. Proudfoot on hyperkahler quotients (HP). One interesting feature of their article is that they consider integrals of equivariant forms over non-compact symplectic manifolds which do not converge, so they formally define these integrals as sums over the zeroes of vector fields, as in the Berline- Vergne localization formula. In this article we introduce a geometric-analytic regularization technique which makes such integrals converge and utilizes the symplectic structure of the manifold. We also prove that the Berline-Vergne localization formula holds for these inte- grals as well. The key step here is to redefine the collection of integrals R M �(X), X 2 g, as a distribution on the Lie algebra g. We expect our regularization technique to generalize to non-compact group actions extending the results of (L1, L2).