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 $S\subset G$).

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

A reference: Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92, Corollary 3.11.

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.

See also: Connectedness of centralizers of semisimple Lie-algebra elements under the action of a semisimple algebraic group

  • $\begingroup$ 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. $\endgroup$ – spin Mar 29 '20 at 9:19
  • $\begingroup$ @spin: I think it is an erroneous reference... $\endgroup$ – Mikhail Borovoi Mar 29 '20 at 9:55

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