Abstract
There are two formulations of the Duistermaat–Heckman theorem, which is the main result of Heckman’s PhD thesis and is presented in [1]. The first (which comes from the original article [1]) describes how the Liouville measure of a symplectic quotient varies. The second describes an oscillatory integral over a symplectic manifold equipped with a Hamiltonian group action and can be characterized by the slogan “Stationary phase is exact”.
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Notes
- 1.
Here, the roots of G are the nonzero weights of its complexified adjoint action. We fix the convention that weights \(\beta \in \mathbf{t}^*\) satisfy \(\beta \in \mathrm{Hom} (\Lambda ^I,{\mathbb Z})\) rather than \(\beta \in \mathrm{Hom} (\Lambda ^I, 2 \pi {\mathbb Z})\) (where \(\Lambda ^I = \mathrm{Ker}(\exp : \mathbf{t}\rightarrow T)\) is the integer lattice). This definition of roots differs by a factor of \(2 \pi \) from the definition used in [10].
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Dwivedi, S., Herman, J., Jeffrey, L.C., van den Hurk, T. (2019). The Duistermaat–Heckman Theorem. In: Hamiltonian Group Actions and Equivariant Cohomology. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-27227-2_10
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