A twisted moment map and its equivariance

TL;DR

This paper constructs explicit symplectic isomorphisms between twisted cotangent bundles of complex flag varieties and coadjoint orbits, demonstrating their $G$-equivariance, thus advancing understanding of geometric structures in Lie theory.

Contribution

It provides explicit $G$-equivariant symplectic isomorphisms involving twisted cotangent bundles and coadjoint orbits, with transition functions given by affine transformations.

Findings

Explicit symplectic isomorphisms constructed

Transition functions are affine transformations

Isomorphisms are $G$-equivariant

Abstract

Let $G$ be a linear connected complex reductive Lie group. The purpose of this paper is to give explicit symplectic isomorphisms from twisted cotangent bundles of the complex generalized flag varieties, whose transition functions are given by affine transformations instead of linear transformations, onto the complex coadjoint semisimple orbits. Moreover, the isomorphisms are shown to be $G$-equivariant.