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Geometry and Physics: Volume IA Festschrift in honour of Nigel Hitchin$
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Andrew Dancer, Jørgen Ellegaard Andersen, and Oscar García-Prada

Print publication date: 2018

Print ISBN-13: 9780198802013

Published to Oxford Scholarship Online: December 2018

DOI: 10.1093/oso/9780198802013.001.0001

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Hyperfunctions, the Duistermaat–Heckman Theorem and Loop Groups

Hyperfunctions, the Duistermaat–Heckman Theorem and Loop Groups

(p.319) 12 Hyperfunctions, the Duistermaat–Heckman Theorem and Loop Groups
Geometry and Physics: Volume I

Lisa C. Jeffrey

James A. Mracek

Oxford University Press

This chapter investigates the Duistermaat–Heckman theorem using the theory of hyperfunctions. In applications involving Hamiltonian torus actions on infinite-dimensional manifolds, the more general theory seems to be necessary in order to accommodate the existence of the infinite-order differential operators which arise from the isotropy representations on the tangent spaces to fixed points. The chapter quickly reviews the theory of hyperfunctions and their Fourier transforms. It then applies this theory to construct a hyperfunction analogue of the Duistermaat–Heckman distribution. The main goal will be to study the Duistermaat–Heckman distribution of the loop space of SU(2) but it will also characterize the singular locus of the moment map for the Hamiltonian action of T×S1 on the loop space of G. The main goal of this chapter is to present a Duistermaat–Heckman hyperfunction arising from a Hamiltonian group action on an infinite-dimensional manifold.

Keywords:   hyperfunction, Hamiltonian group action, Duistermaat–Heckman theorem, Fourier transform infinite-dimensional manifold

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