2 Changed to more restricted statement.

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 direct reference to your exact question, at least off-hand, but I claim that the following in more limited statement is true:

Theorem: A compact connected Lie group G has the property (**) "For every closed subgroup 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 it in particular holds for all elementary abelian p-subgroups $H$ for all p. But this implies that , 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: By using that decending chains of connected subgroups in a compact Lie group have to eventually stabilize, it is enough to check that (**) holds for finite abelian subgroups $H$. So to finish the proof, we 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.

1

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.

I don't know a direct reference to your exact question, at least off-hand, but I claim that the following in true:

Theorem: A compact connected Lie group G has the property (**) "For every closed 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 it in particular holds for all elementary abelian p-subgroups for all p. But this implies that 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: By using that decending chains of connected subgroups in a compact Lie group have to eventually stabilize, it is enough to check that (**) holds for finite abelian subgroups $H$. So to finish the proof, 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.

--

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.