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Jordan decompositions in Lie algebras and their duals

Loren Spice, Cheng-Chiang Tsai

TL;DR

This work provides a uniform treatment of Jordan decompositions in the Lie algebra $\mathfrak{g}$ and its dual $\mathfrak{g}^*$ for connected reductive groups, proving existence in both settings and revealing non-uniqueness on $\mathfrak{g}^*$ with a precise upper bound. It reduces the central questions to two characteristic-$2$ families (adjoint of $\mathrm{Sp}_{2n}$ and coadjoint of $\mathrm{SO}_{2n+1}$) and establishes Chevalley-restriction-type results via two invariant-theoretic approaches, including a universal homeomorphism relating semisimple quotients. The paper provides explicit counterexamples to non-uniqueness, refines KW76’s statements, and presents a sharpening (Theorem $\mathrm{uni}$) that isolates all possible sources of non-uniqueness, while showing uniqueness for semisimple and nilpotent elements. Collectively, these results enhance the understanding of adjoint and coadjoint actions, clarify the structure of GIT quotients, and supply uniform, characteristic-aware tools for invariant theory in Lie theory.

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 $G$.

Jordan decompositions in Lie algebras and their duals

TL;DR

This work provides a uniform treatment of Jordan decompositions in the Lie algebra and its dual for connected reductive groups, proving existence in both settings and revealing non-uniqueness on with a precise upper bound. It reduces the central questions to two characteristic- families (adjoint of and coadjoint of ) and establishes Chevalley-restriction-type results via two invariant-theoretic approaches, including a universal homeomorphism relating semisimple quotients. The paper provides explicit counterexamples to non-uniqueness, refines KW76’s statements, and presents a sharpening (Theorem ) that isolates all possible sources of non-uniqueness, while showing uniqueness for semisimple and nilpotent elements. Collectively, these results enhance the understanding of adjoint and coadjoint actions, clarify the structure of GIT quotients, and supply uniform, characteristic-aware tools for invariant theory in Lie theory.

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 .
Paper Structure (9 sections, 9 theorems, 11 equations)

This paper contains 9 sections, 9 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Notation and preparation
  3. Remarks on Jordan decompositions
  4. Reduction to $\mathfrak{sp}_{2n}$ and $\mathfrak{so}_{2n+1}^*$
  5. First approach
  6. Second approach
  7. Non-uniqueness of the Jordan decomposition
  8. Uniqueness of the Jordan decomposition
  9. Uniqueness of Jordan decompositions for semisimple and nilpotent elements

Key Result

Theorem 1.1

Let $G$ be a connected reductive group over an algebraically closed field $k$. Write $\mathfrak{g}$ and $\mathfrak{g}^*$ for the Lie algebra and dual Lie algebra of $G$, or the associated vector groups. When $V$ is an affine variety on which $G$ acts, we write $V/\!/ G$ for the GIT quotient $\operat

Theorems & Definitions (26)

  • Theorem 1.1
  • Conjecture 1.2
  • Remark 2.1
  • Lemma 2.2
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Remark 7.1
  • ...and 16 more