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I would be grateful if someone could provide a reference/proof of the following fact (or give a counterexample if I've misunderstood and it's false!)

Let $G$ be a reductive group over a field $F$ (in practice I really only care about local and global fields). Suppose that $G$ contains an elliptic maximal $F$-torus (i.e. a maximal torus defined over $F$ that is anisotropic modulo $Z(G)$ ). Suppose that $\gamma \in G(F)$ is an elliptic element (i.e. the identity component of the center of the identity component of the centralizer of $\gamma$ in $G$ is an elliptic torus).

Then, fact: $\gamma$ is contained in an elliptic maximal torus.

This is alluded to on page 392 of Kottwitz's Stable Trace Formula: Elliptic Singular Terms, but I would appreciate a reference that proves this fact, or at least confirmation that I am not misunderstanding this passage.

Thanks!

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    $\begingroup$ By Theorem 6.21 of Platonov-Rapinchuk (algebraic groups and number theory), any reductive group over a non archimedean local field admits a maximal elliptic torus. Now let $G_\gamma$ be the centralizer of $\gamma$ and $T$ a maximal elliptic torus in $G_\gamma$. Then $T$ should be a maximal elliptic torus in $G$. I do not know any reference for the global case. $\endgroup$ – Paul Broussous Jun 12 at 15:33
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    $\begingroup$ Replace everywhere "maximal elliptic" by "elliptic maximal" in my previous comment. $\endgroup$ – Paul Broussous Jun 12 at 15:44
  • $\begingroup$ Thanks for this. I guess the Archimedean case is particularly interesting, as there one is not guaranteed an elliptic maximal torus so one would need to modify the argument. $\endgroup$ – Alexander Jun 12 at 16:19
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    $\begingroup$ @LSpice Indeed I implicitely assumed $\gamma$ semisimple. $\endgroup$ – Paul Broussous Jun 13 at 8:38
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    $\begingroup$ Oh, I see. I missed that you said "$Z(C_G(\gamma)^\circ)^\circ$ elliptic" rather than "$C_G(\gamma)^\circ$ elliptic". $\endgroup$ – LSpice Jun 13 at 12:37

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