Generically free representations III: exceptionally bad characteristic
In parts I and II, we determined which 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. In a later work, we combine these results with those of Guralnick--Lawther--Liebeck to show 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.