The Asymptotic Behavior of Invariant Collective Motion

Let G be a connected complex reductive group acting on a smooth algebraic variety X. Then the cotangent bundle T ∗ X on X carries a canonical symplectic structure, and the G-action induces a moment map Φ : T ∗ X → g . Consider the Hamiltonian vector fields attached to functions of the form f ◦ Φ with f ∈ C[g]. In this paper we study the asymptotic behavior of the associated flow (a so-called invariant collective motion) and show that it possesses a symmetry with respect to a finite reflection group WX . This is applied to the theory of equivariant embeddings of X. The approach is purely algebraic. More specifically: Choose any generic point α ∈ T ∗ X . Because the functions f ◦Φ with f ∈ C[g] are in involution (i.e., their Poisson product vanishes), the flow through α is in the orbit of an abelian group Aα. It is known (see [GS]) that this orbit is also the orbit for the connected isotropy group GΦ(α). This implies that Aα is a linear algebraic group and it turns out that it is a torus. The projection of this orbit to X is called a flat of X and just equals GΦ(α)π(α). In case, X is the complexification of a symmetric space, a flat in our sense is the complexification of a usual flat (=maximal totally geodesic, flat submanifold). Let X ⊆ X be a normal equivariant embedding. The main point of this paper is to study the closure of a generic flat in X. This will be done in two different steps. The first one is to show that a certain finite group WX acts on them. Consider the family of tori α 7→ Aα. Although every two of these groups are isomorphic to each other, the family cannot in general be trivialized globally. But we show that it can be trivialized on a finite cover T̂X of an open subset of T ∗ X . Hence, there is an action of a torus AX