The Asymptotic Behavior of Invariant Collective Motion

Let G be reductive and X a smooth G-variety. Then the cotangent bundle T_X^* carries a symplectic structure and the G-action gives rise to a moment map T_X^*-->g^* (with g=Lie G). Let f be a regular function on T_X^* which is induced by an Ad G-invariant function on g^*. The associated Hamiltonian flow is called invariant collective. In this paper we prove that the invariant collective flow is symmetric under the little Weyl group W_X of X.

The main application is: Let Z(X) be the set of all G-invariant valuations of k(X) which are trivial on k(X)^B, B=Borel subgroup. Then Z(X) is canonically in bijection to a Weyl chamber of W_X.

Appeared in: Inventiones mathematicae 116 (1994) 309-328

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Last updated: August 25, 2006