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For a semisimple complex Lie algebra $\mathfrak{g}$ and a regular element $X\in \mathfrak g$ the centralizer of $X$ in $\mathfrak g$ is a Cartan subalgebra (see Knapp, 'Lie Groups beyond an introduction', Thm. 2.9'). I'm wondering whether something similar holds on the group level. More precisely, if $G$ is a real semisimple linear group (I'm mostly interested in $\operatorname{Sp}(2n,\mathbb R)$), for which $x\in G$ does it hold that the centralizer of $x$ in $G$ is a Cartan subgroup?

Each regular element $x\in G$ is contained in some Cartan subgroup $H$ which is abelian (Knapp, Thm. 7.108). Hence, $H\subseteq Z_G(x)$. Can one impose additional conditions on $x$ to force equality?

In the example $G=\operatorname{SL}(2,\mathbb C)/\{\pm 1\}$ on page 427 one sees that the centralizer of the diagonal matrix $x$ with entries $z$ and $z^{-1}$ is a Cartan subgroup iff $z\neq \pm 1$ and $z\neq \pm i$ where $z\neq \pm 1$ is just regularity. In particular, the centralizer is a Cartan subgroup iff $x$ and $x^2$ are regular, and regularity alone is not sufficient.

If $G$ is complex, then $H$ is of finite index in $Z_G(x)$ (Knapp, Cor. 7.106) but I didn't find a condition for equality.

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The condition is known as "strongly regular" (and I think that the modern usage is "regular semisimple", hence also "strongly regular semisimple", not just "regular", which is often understood to allow such things as regular unipotents). By Steinberg - Regular elements of semisimple algebraic groups, say §§2.14, 2.15, the condition for a regular semisimple element $x$ to be strongly regular semisimple is that the subgroup of the Weyl group $W(G, Z_G(x)^\circ)$ that fixes $x$ is trivial; and this extra condition is vacuous if $G$ is simply connected.

(Incidentally, in positive characteristic, a similar subtlety with possibly disconnected centralisers can arise even for regular semisimple elements of the Lie algebra. For example, in characteristic $2$, the centraliser in $\operatorname{PGL}_2$ of the element $\operatorname{diag}(1, 0) \in \mathfrak{pgl}_2$ is the full normaliser of a torus.)

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  • $\begingroup$ Thanks! I wonder whether this is related to the notion of hyperregularity in this paper jstor.org/stable/1970790 $\endgroup$
    – yolassr
    Commented Dec 9 at 12:06
  • $\begingroup$ @yolassr, re, I don't think so. Strong regularity is sensitive to isogeny; the element $\operatorname{diag}(i, -i)$ of $\operatorname{SL}_2$ is strongly regular semisimple, but its image in $\operatorname{PGL}_2$ is not. More generally, every lift to $G_\text{sc}$ of an rss element $\gamma$ of $G$ is srss. On the other hand, as far as I can tell, hyperregularity of an element $\gamma$ depends only on the action on $\bigwedge\operatorname{Lie}(G)$, hence only on the image of $\gamma$ in $G/{\operatorname Z(G)}$. $\endgroup$
    – LSpice
    Commented Dec 9 at 17:11

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