  In this article we investigate the Duistermaat-Heckman theorem using the
theory of hyperfunctions. In applications involving Hamiltonian torus actions
on infinite dimensional manifolds, this more general theory seems to be
necessary in order to accomodate the existence of the infinite order
differential operators which arise from the isotropy representations on the
tangent spaces to fixed points. We will quickly review of the theory of
hyperfunctions and their Fourier transforms. We will then apply this theory to
construct a hyperfunction analogue of the Duistermaat-Heckman distribution. Our
main goal will be to study the Duistermaat-Heckman hyperfunction of $\Omega
SU(2)$, but in getting to this goal we will also characterize the singular
locus of the moment map for the Hamiltonian action of $T\times S^1$ on $\Omega
G$. The main goal of this paper is to present a Duistermaat-Heckman
hyperfunction arising from a Hamiltonian action on an infinite dimensional
manifold.
