Generically free representations III: extremely bad characteristic

TL;DR

This paper completes the classification of generically free representations of simple algebraic groups in extremely bad characteristic, addressing complex cases where Lie algebra structure complicates previous dimension-based methods.

Contribution

It provides a comprehensive analysis of generically free representations in the challenging setting of extremely bad characteristic, extending prior classifications.

Findings

Complete classification of generically free representations in bad characteristic

Identification of new cases where Lie algebra structure is more complex

Extension of previous results to more difficult algebraic group cases

Abstract

In parts I and II, we determined which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for Lie($G$), i.e., which $V$ have an open subset consisting of vectors whose stabilizer in Lie($G$) is zero, with some assumptions on the characteristic of the field. This paper settles the remaining cases, which are of a different nature because Lie($G$) has a more complicated structure and there need not exist general dimension bounds of the sort that exist in good characteristic.