Jordan decompositions in Lie algebras and their duals

TL;DR

This paper explores Jordan decompositions in Lie algebras and their duals, highlighting their non-uniqueness and providing bounds, along with Chevalley-restriction results for GIT quotients in reductive groups.

Contribution

It offers a uniform discussion of Jordan decompositions in Lie algebras and duals, presents a counterexample to their uniqueness, and establishes bounds and GIT quotient properties.

Findings

Counterexample to uniqueness of Jordan decompositions on dual Lie algebras

Upper bound on non-uniqueness of Jordan decompositions

Chevalley-restriction-type results for GIT quotients

Abstract

We provide a discussion of Jordan decompositions in the Lie algebra, and the dual Lie algebra, of a reductive group in as uniform a way as possible. We give a counterexample to the claim that Jordan decompositions on the dual Lie algebra are unique, and state an upper bound on how non-unique they can be. We also prove some Chevalley-restriction-type claims about GIT quotients for the adjoint and co-adjoint actions of $G$.