Lifting $mathfrak{sl}_2$-triples


  • $k$ be an algebraically closed field,
  • $G$ a (smooth, connected) reductive algebraic group over $k$,
  • $H$ a (smooth, connected) reductive group of semisimple rank 1, and
  • $T$ a maximal torus in $H$.

I am specifically interested in the case where the characteristic of $k$ is a bad prime for $G$.

Suppose that we are given a group embedding $F_T : T to G$ and a $T$-equivariant Lie-algebra embedding $f_H : mathfrak h to mathfrak g$, such that the restrictions to $mathfrak t$ of $f_H$ and the derivative of $F_T$ agree. Can we extend $F_T$ to an embedding $H to G$ whose derivative is $f_H$?

(I know that most results in this area are stated under the assumption of good characteristic, or even of characteristic 0, but I don’t actually know any of the counterexamples or where to look for them, so I can’t tell if the hypothesis of the existence of $T$ is strong enough to allow me to overcome bad characteristic.)