We prove a myriad of results related to the stabilizer in an algebraic group G of a generic vector in a representation V of G over an algebraically closed field k. Our results are on the level of group schemes, which carries more information than considering both the Lie algebra of G and the group of k-points. For G simple and V faithful and irreducible, we prove the existence of a stabilizer in general position, sometimes called a principal orbit type. We determine those G and V for which the stabilizer in general position is smooth, or , or there is a whose stabilizer in G is trivial.
To Gopal Prasad, in honor of his 75th birthday
"Generic Stabilizers for Simple Algebraic Groups." Michigan Math. J. 72 343 - 387, August 2022. https://doi.org/10.1307/mmj/20217216