# Maximal split tori in quasisplit groups

Let $$k$$ be a number field. Let $$G$$ be a quasisplit (but not split) semisimple group over $$k$$. Let $$S$$ be a maximal $$k$$-split torus in $$G$$. Let $$T$$ be the centralizer $$Z_G(S)$$ of $$S$$; it is a maximal torus in $$G$$. Does $$S$$ remain a maximal split torus over nonarchimedean completions $$k_v$$?

It is certainly not true if $$G$$ is not necessarily quasisplit, for example if $$G$$ is an inner form of a split group, like $$G=SL_n(D)$$.

(This may be obvious but I also don't know much about the splitting behavior of outer forms of split groups.)

• My example was meant to be an inner form of SL$_n$ - that's not quasisplit. I edited for clarity. – Grad student Jan 20 at 22:11
• Thanks, this was what I was unsure about (I don't see from a glance which forms are quasisplit). – YCor Jan 20 at 22:24
• It can happen that $G$ splits over $k_v$ (consider, for instance, the quasi-split $SU(2)$ taken relative to a quadratic extension $K/k$ such that $K\otimes_k k_v\cong k_v\times k_v$). In this case, $S$ will not be a maximal split torus of $G$ over $k_v$. In fact, if I recall correctly, any $G$ should split over almost all completions. – Uriya First Jan 21 at 7:04
• For special unitary groups, the splitting behaviour of the torus is determined by the splitting behaviour of the place in the corresponding quadratic extension. Thus such G will split over half of all completions. – Peter McNamara Jan 21 at 11:24
• Probably it's helpful to consult the list of Lie types at the end of the article by Tits mathscinet.ams.org/mathscinet-getitem?mr=0224710 – Jim Humphreys Jan 21 at 19:53