# Connectedness of the stabilizer in a semisimple group of a semisimple element in the Lie algebra: a reference request

Let $$G$$ be a (connected) semisimple algebraic group over an algebraically closed field $$k$$ of characteristic 0. We consider the adjoint representation $${\rm Ad}\colon G\to {\rm GL}({\mathfrak g}),$$ where $${\mathfrak g}={\rm Lie}\ G$$.

I am looking for a reference to a proof of the following assertion:

Proposition. For any semisimple element $$X\in{\mathfrak g}$$, its stabilizer $${\rm Stab}_G(X)\subset G$$ with respect to the adjoint representation is connected.

I think that I can prove the proposition; see my proof below.

Proof. Our semisimple element $$X$$ is contained in a Cartan subalgebra $${\mathfrak t}$$ of $${\mathfrak g}$$, which is the Lie algebra of a maximal torus $$T$$ of $$G$$. Then $${\mathfrak t}$$ is an algebraic subalgebra of $${\mathfrak g}$$. Let $${\langle X\rangle_{\rm alg}}\subset {\mathfrak g}$$ denote the smallest algebraic subalgebra of $${\mathfrak g}$$ containing $$X$$; then $${\langle X\rangle_{\rm alg}}\subseteq {\mathfrak t}$$. It follows that $${\langle X\rangle_{\rm alg}}={\rm Lie}\ S$$ for some subtorus $$S\subseteq T$$. Now (in characteristic 0) we have $${\rm Stab}_G(X)=\bigcap_{Y\in {\langle X\rangle_{\rm alg}}} {\rm Stab}_G(Y)=C_G(S),$$ where $$C_G(S)$$ denotes the centralizer of the torus $$S$$ in $$G$$. By Theorem 22.3 of Humphreys' book "Linear Algebraic Groups", $$C_G(S)$$ is connected, as required.

Edit: A similar argument shows that for any commutative subalgebra $${\mathfrak a}\subset {\mathfrak g}$$ consisting of semisimple elements, its centralizer in $$G$$ $$C_G({\mathfrak a}):=\bigcap_{X\in {\mathfrak a}} {\rm Stab}_G(X)$$ is connected (because the "algebraic closure" $$\langle {\mathfrak a}\rangle_{\rm alg}$$ of $$\mathfrak a$$ is the Lie algebra of some torus $$S\subset G$$).

• Isn't it a classic result which is in any book on algebraic groups (say, Borel). – user6976 Mar 27 '20 at 17:16

In positive characteristic $$p$$: see loc. cit., Theorem 3.14. It says that if (and only if) $$p$$ is not a torsion prime for $$G$$, then $$C_G({\mathfrak a})$$ is connected for any commutative subalgebra $${\mathfrak a}\subset {\mathfrak g}$$ consisting of semisimple elements.
• Steinberg says there is a different proof for $k = \mathbb{C}$, and gives reference to "Lemma 5" in Volume 2 of Seminaire Chevalley. I do not find such a lemma there, and I wonder what is the intended reference. – spin Mar 29 '20 at 9:19