# 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. Commented Jun 12, 2019 at 15:33
• Replace everywhere "maximal elliptic" by "elliptic maximal" in my previous comment. Commented Jun 12, 2019 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. Commented Jun 12, 2019 at 16:19
• @LSpice Indeed I implicitely assumed $\gamma$ semisimple. Commented Jun 13, 2019 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". Commented Jun 13, 2019 at 12:37