Lie Algebra Decomposition Classes for Reductive Algebraic Groups in Arbitrary Characteristic

TL;DR

This paper studies the structure of Lie algebra decomposition classes for reductive algebraic groups over algebraically closed fields of any characteristic, extending previous results and introducing new concepts to handle bad characteristic cases.

Contribution

It extends the theory of decomposition classes to arbitrary characteristic and introduces Levi-type classes to address challenges in bad characteristic.

Findings

Extended decomposition class results to arbitrary characteristic

Introduced Levi-type decomposition classes for bad characteristic

Extended Lusztig-Spaltenstein induction properties

Abstract

Decomposition classes provide a way of partitioning the Lie algebras of an algebraic group into equivalence classes based on the Jordan decomposition. In this paper, we investigate the decomposition classes of the Lie algebras of connected reductive algebraic groups, over algebraically closed fields of arbitrary characteristic. We extend some results previously proved under restrictions on the characteristic, and introduce Levi-type decomposition classes to account for some of the difficulties encountered in bad characteristic. We also establish properties of Lusztig-Spaltenstein induction of non-nilpotent orbits, extending the known results for nilpotent orbits.