# center of the centralizer of semisimple element

Let $G$ be an adjoint group over an algebraically closed field $k$ and $s\in G$ a semisimple element.

Let $H=C_{G}(s)^{0}$ the neutral component of the centralizer of $s$. Do we have that the center of $H$ is itself connected?

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No, we do not have that. Take $G=PSO_8$, take for $s$ the image of ${\rm diag}(1,1,1,1,-1,-1,-1,-1)\in SO_8$, then $H=(SO_4\times SO_4)/\mu_2$, and the center of $H$ is isomorphic to $\mu_2$, hence not connected. –  Mikhail Borovoi Jan 8 '13 at 20:30
The general study of this and related phenomena is "Borel-de Siebenthal theory". –  Allen Knutson Jan 8 '13 at 20:47

{EDIT] Since I was discussing a slightly different question than the one actually asked, I'll start over with your connected group $H$ and look at its center.

The question illustrates some of the fine points in the Borel-Tits structure theory of reductive groups, which are worth considering beyond Mikhail's counterexample (which I guess has to require good characteristic, here different from 2.) In the formulation of the question, I'm assuming that $G$ is meant to be connected as well as semisimple of adjoint type over an arbitrary algebraically closed field. Here the general result for a semisimple element $s \in G$ is that the centralizer is reductive but not necessarily connected. In particular, your group $H$ is both connected and reductive.

More precisely, Borel-Tits show (in greater generality) that the centralizer $C_G(s)$ of $s$ is generated by some connected subgroups of $G$ (including a maximal torus) along with perhaps part of the Weyl group. Moreover, those connected subgroups are enough to generate the identity component $H$, which contains all unipotent elements of the centralizer (a relevant issue in characteristic $p>0$).

I think the basic problem you encounter is that while $G$ itself is of adjoint type, there is no reason why the connected semisimple derived group of $H$ should also be of adjoint type. So the center of $H$ might well be the direct product of the nontrivial finite center of this derived group and a torus. Beyond Mikhail's proposed example, I expect that a transparent example might be seen when $G$ has type $G_2$: this group is both simply connected and of adjoint type, but will contain subgroups isomorphic to $\mathrm{SL}_2$ that aren't of adjoint type. This might arise as the derived group of a larger connected centralizer in which it's a direct factor. (But whether this configuration actually occurs I'd have to check more carefully.)

A concrete example in rank 2 comes from the Carter-Deriziotis theory discussed at the end of Chapter 2 in my book on conjugacy classes. For $G$ still of type $G_2$, there is a connected reductive (actually semisimple) subgroup $H$ isomorphic to $\mathrm{SL}_3$ which by their criterion (in good characteristic) is the centralizer of some semisimple element. (Here $G$ is simply connected, so in fact all centralizers of semisimple elements are connected, by Springer-Steinberg.) So the center of $H$ is disconnected. (In Deriziotis' version the field has prime characteristic, but the result is similar to that of Borel and deSiebenthal in characteristic 0 mentioned by Allen: it relies on the extended Dynkin diagram.)

P.S. Extending this general theory, John Kurtzke looked more closely at the centers of centralizers; his second paper corrects some points in the first. He showed for instance that when $s$ is irregular, its centralizer is typically disconnected. He also bounded the dimensions of centers of centralizers, but didn't discuss their connectedness directly.

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