Let


* $k$ 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.)



Lifting $\mathfrak{sl}_2$-triples
Typically not, even for good characteristic. This comes up indirectly in my paper with Adam Thomas [The Jacobson–Morozov theorem and complete reducibility of Lie subalgebras, JLMS](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms.12067) [arxiv version](https://arxiv.org/abs/1507.06234).


The nilpotent orbits of $G$ on $\mathfrak g$ in characteristic $0$ have representatives $e\_{\mathbb Z}$ which are sums of root elements and which remain in distinct orbits when base-changed to a field of good characteristic. Furthermore, each can be completed to an $\mathfrak{sl}\_2$-triple $(e\_{\mathbb Z},h\_{\mathbb Z},f\_{\mathbb Z})\in\mathfrak{g}\_{\mathbb Z}^3$—this is a result of Pommerening, though Thomas and I extend it slightly. Denote by $(e,h,f)$ the elements $(e\otimes 1,h\otimes 1,f\otimes 1)$ of $(\mathfrak{g}\_{\mathbb Z}\otimes k)^3$. We have $h$ is toral, so $\mathfrak t=\langle h\rangle$ is a torus of $\mathfrak{g}$. Since $h$ can be expressed as an integral combination of coroots, $\mathfrak t$ can be lifted to a torus $T$ such that $T$ normalises $\langle e\rangle$ and $\langle f\rangle$. Let $\mathfrak{h}=\langle e,h,f\rangle\cong \mathfrak{sl}\_2$. We have that $T$ normalises $\mathfrak{h}$, so that the embedding $\mathfrak{h}\to\mathfrak{g}$ is $T$ equivariant. Suppose we could lift this to $F\_H:H\to G$ for some reductive $H$. (So $H\cong \operatorname{SL}\_2$ or $\operatorname{PGL}\_2$.)


Take the case that $e$ is a regular nilpotent element of $\mathfrak g$. Then a non-trivial unipotent element of $H$ must be regular too. But for $p$ less than the Coxeter number of $G$, a regular unipotent element has order at least $p^2$, so cannot be embedded into an $SL\_2$ or $PGL\_2$ (whose unipotents have order $p$).


EDITED: I initially expected this problem goes away if you insist $p$ is bigger than the Coxeter number of $G$, *because* I said there is a 1-1 correspondence between conjugacy classes of $\mathfrak{sl}\_2$-subalgebras of $\mathfrak g$ and subgroups of $G$ of type $A\_1$. This is obviously wrong however, since in positive characteristic one gets many more irreducible representations for $SL\_2$ than for $\mathfrak{sl}\_2$, giving many more $SL\_2$ or $PGL\_2$ subgroups of $GL\_n$ than $\mathfrak{sl}\_2$ subalgebras, c.f. Steinberg's tensor product theorem.


By way of correction, if $p$ is bigger than the Coxeter number, all unipotent elements have order $p$. Then one can cite Theorem 4.2 of
*Lawther, R.; Testerman, D. M.*, [**(A\_1) subgroups of exceptional algebraic groups**](http://dx.doi.org/10.1090/memo/0674), Mem. Am. Math. Soc. 674, 131 p. (1999). [ZBL0936.20039](https://zbmath.org/?q=an:0936.20039). in which an $A\_1$ subgroup is guaranteed such that the Lie algebra of its root group contains nilpotent elements of the same Bala–Carter type; *a fortiori* one that corresponds under a Springer isomorphism.


