TITLE: When are all centralizers in a Lie group connected?
QUESTION [15 upvotes]: Let $G$ be a compact connected Lie group acting on itself by conjugation,
$$ G\times G\to G,\qquad (\sigma,h)\mapsto \sigma h \sigma^{-1}.$$
The fixed point set of a closed subgroup $H\le G$ equals the centralizer $C_G(H)$ of $H$ in $G$.
For a current project I am searching for examples of actions $G\times X\to X$ such that every fixed point set $X^H$ is connected, and I would like to fit the above conjugation action into this framework. Hence my questions:

Are there any conditions on a connected Lie group $G$ which ensure that the centralizer $C_G(H)$ of every closed subgroup $H\le G$ is connected?
Can someone give examples where this is true? I was thinking the unitary groups $U(n)$ might be candidates, since at least their centre $U(1)$ (which clearly is contained in every centralizer) is connected.

Thanks, and apologies if this is too vague or elementary.

REPLY [6 votes]: Centralizers of various types of subgroups, and their relationship to invariants like the cohomology of the classifying space $BG$ has a long history of study, at least going back to the papers from the 50s by Borel and others.
EDIT: Changed what's below to a more limited statement.
I don't know a reference to your exact question, but the following more limited statement is true:
Theorem: A compact connected Lie group G has the property (**) "For every elementary abelian $p$-subgroup $H \leq G$, $C_G(H)$ is connected" if and only if $G$ has torsion-free fundamental group and has a finite cover isomorphic to a product of copies of $SU(n)$, $Sp(n)$, $S^1$ for various $n \geq 1$.
In particular, if you in addition assume $G$ to be simply connected, the only examples are products of $SU(n)$ and $Sp(n)$ for various $n$.
Sketch of proof:
1) The condition is necessary: If (**) holds holds for all elementary abelian p-subgroups $H$ for all p, then every elementary abelian p-subgroup has to be contained in a maximal torus. By a result of Borel this implies that the fundamental group is torsion free. One easily checks case-by-case that the exceptional groups and Spin(n), $n \geq 6$ cannot be involved in $G$ since they have non-toral elementary abelian p-subgroup (which remain so even after modding out by a central subgroup). This leaves the cases above.
2) The condition is sufficient: We just have to check that the centralizer of an element of finite order of a group as above will again be a group of the same form, which I think you can do by hand.
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Further info: The groups in the list above are exactly the ones whose classifying spaces have integral cohomology rings a finitely generated polynomial algebra. In fact any polynomial ring which occur as the cohomology ring of the space has to be of this form ---  this is the so-called Steenrod problem, whose solution is described in Andersen-Grodal: The Steenrod problem of realizing polynomial cohomology rings. J. Topology 1 (2008), 747-760. (DOI: 10.1112/jtopol/jtn021). 
Also, see Andersen-Grodal-Møller-Viruel The classification of p-compact groups for p odd. Annals of Math. 167 (2008), 95--210, in particular section 10+12 for a summary of the relationship between subgroups of $G$, their centralizers, and cohomology. Our setup is more general in that paper (we study p-compact groups), but it contains references to the papers of Borel etc.