The Hesselink stratification of nullcones and base change

TL;DR

This paper provides a uniform, case-free proof of Lusztig's conjectures on unipotent pieces in algebraic groups, extending Dynkin--Kostant theory to all characteristics and including results for Lie algebra actions.

Contribution

It offers a case-free proof of Lusztig's conjectures on unipotent pieces, extending the theory uniformly across all characteristics.

Findings

Proves Lusztig's conjectures on unipotent pieces without case restrictions.

Extends Dynkin--Kostant theory to positive characteristic.

Provides analogous results for Lie algebra and coadjoint actions.

Abstract

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic $p \ge 0$. We give a case-free proof of Lusztig's conjectures [Unipotent elements in small characteristic, {\em Transform. Groups} 10 (2005), 449--487] on so-called unipotent pieces. This presents a uniform picture of the unipotent elements of $G$ which can be viewed as an extension of the Dynkin--Kostant theory, but is valid without restriction on $p$. We also obtain analogous results for the adjoint action of $G$ on its Lie algebra $\gl$ and the coadjoint action of $G$ on $\gl^*$.