Equivariant symplectic geometry of cotangent bundles. II

We examine the structure of the cotangent bundle $T^*X$ of an algebraic variety X acted on by a reductive group $G$ from the viewpoint of equivariant symplectic geometry. In particular, we construct an equivariant symplectic covering of $T^*X$ by the cotangent bundle of a certain variety of horospheres in $X$, and integrate the invariant collective motion on $T^*X$. These results are based on a “local structure theorem” describing the action of a certain parabolic in $G$ on an open subset of $X$, which is interesting by itself.