The Duistermaat-Heckman integration formula on generalized flag manifolds

Andreas Arvanitoyeorgos


Abstract. Let $M$ be a $2n$-dimensional symplectic manifold with symplectic 2-form $\omega$. The partition function of a classical system whose dynamics is governed by the Hamiltonian function $H$ can be written as an integral $\int _Me ^{i\phi H}\frac{\omega ^n}{n!}$ ($\phi\in\R$). The Duistermaat-Heckman integration formula states that if the Hamiltonian vector field has only isolated zeros and it is almost periodic, then the previous integral can be expressed as a certain sum over the critical points of the Hamiltonian. We will study this integration formula for a class of K\"ahler manifolds, the generalized flag manifolds. These are homogeneous spaces of the form $G/C(T)$, where $G$ is a compact Lie group and $T$ a maximal torus in $G$. Equivalently, they can be expressed as adjoint orbits of the adjoint representation of $G$ in its Lie algebra. In our approach we use well known results of R. Bott and H. Samelson about the Morse theory of height functions on an adjoint orbit, to give a Lie algebraic expression for the Duistermaat-Heckman formula, suitable for calculations.

AMSclassification. Primary 53C80, 53C30, 53C21, 58F05; Secondary 57R70, 55N91

Keywords. Generalized flag manifolds, Duistermaat-Heckman formula, Morse theory, Homogeneous spaces, Equivariant cohomology