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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.)

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  • $\begingroup$ My example was meant to be an inner form of SL$_n$ - that's not quasisplit. I edited for clarity. $\endgroup$
    – Not a grad student
    Commented Jan 20, 2019 at 22:11
  • $\begingroup$ Thanks, this was what I was unsure about (I don't see from a glance which forms are quasisplit). $\endgroup$
    – YCor
    Commented Jan 20, 2019 at 22:24
  • $\begingroup$ 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. $\endgroup$
    – Uriya First
    Commented Jan 21, 2019 at 7:04
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    $\begingroup$ 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. $\endgroup$
    – Peter McNamara
    Commented Jan 21, 2019 at 11:24
  • $\begingroup$ 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 $\endgroup$
    – Jim Humphreys
    Commented Jan 21, 2019 at 19:53

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