Equivariant symplectic geometry of cotangent bundles, II

TL;DR

This paper explores the equivariant symplectic structure of cotangent bundles of algebraic varieties under reductive group actions, introducing a covering by cotangent bundles of horosphere varieties and analyzing invariant motions.

Contribution

It constructs an equivariant symplectic covering of cotangent bundles and integrates invariant collective motion, based on a local structure theorem for group actions.

Findings

Constructed an equivariant symplectic covering of $T^{*}X$

Integrated invariant collective motion on $T^{*}X$

Developed a local structure theorem for group actions

Abstract

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.