# Elliptic Maximal Tori and Elliptic Elements

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!

• 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. – Paul Broussous Jun 12 at 15:33
• Replace everywhere "maximal elliptic" by "elliptic maximal" in my previous comment. – Paul Broussous Jun 12 at 15:44
• 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. – Alexander Jun 12 at 16:19
• @LSpice Indeed I implicitely assumed $\gamma$ semisimple. – Paul Broussous Jun 13 at 8:38
• Oh, I see. I missed that you said "$Z(C_G(\gamma)^\circ)^\circ$ elliptic" rather than "$C_G(\gamma)^\circ$ elliptic". – LSpice Jun 13 at 12:37