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The Duistermaat-Heckman integration formula on generalized
flag manifolds

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*Andreas Arvanitoyeorgos*

**E-mail:** taarv@hol.gr
**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