Generically free representations II: irreducible representations

TL;DR

This paper classifies faithful irreducible representations of simple algebraic groups that are generically free for their Lie algebras, extending known results from characteristic zero to prime characteristic with new proofs and methods.

Contribution

It provides a classification of generically free irreducible representations over arbitrary fields, including prime characteristic, with a shorter proof and handling of new phenomena.

Findings

Classification of generically free representations for simple algebraic groups

Extension of results from characteristic zero to prime characteristic

Use of bounds and computer calculations for remaining cases

Abstract

We determine 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. This relies on bounds on $\dim V$ obtained in prior work (part I), which reduce the problem to a finite number of possibilities for $G$ and highest weights for $V$, but still infinitely many characteristics. The remaining cases are handled individually, some by computer calculation. These results were previously known for fields of characteristic zero, although new phenomena appear in prime characteristic; we provide a shorter proof that gives the result with very mild hypotheses on the characteristic. (The few characteristics not treated here are settled in part III.) These results are related to questions about invariants and the existence of a stabilizer in general position.