Generically free representations II: irreducible representations

Skip Garibaldi, Robert M. Guralnick

Published 2017-11-17Version 1

We determine which faithful irreducible representations $V$ of a simple linear algebraic group $G$ are generically free for $\mathrm{Lie}(G)$, i.e., which $V$ have an open subset consisting of vectors whose stabilizer in $\mathrm{Lie}(G)$ 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; 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.) Combining these results with those of Guralnick--Lawther--Liebeck shows that for any irreducible module for a simple algebraic group, there is a generic stabilizer (as a group scheme) and gives a classification of the generic stabilizers in all cases. These results are also related to questions about invariants.