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Let $k$ be a perfect field.

Recall that an algebraic torus $T$ over $k$ is called quasi-split if there exists some finite étale $k$-algebra $A$ such that $$T \cong \mathrm{R}_{A/k} \mathbb{G}_m.$$

A reductive group $G$ over $k$ is called quasi-split if it contains a Borel subgroup $B$ over $k$.

A priori, I see no reason why these two definitions should be related; Indeed any algebraic torus is quasi-split in the second sense (take $B=T$), but generally not in the first sense. However, I have seen it claimed, or at least implicitly used, in proofs that for semisimple $G$, the latter implies the former. Namely:

Let $G$ be a semisimple algebraic group over $k$ with a maximal torus $T$. If $G$ is quasi-split, then is $T$ quasi-split? Does the converse hold?

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    $\begingroup$ I'm pretty sure that $SU(2, 2)$ over $\mathbf{R}$ is semisimple and quasi-split, but its maximal torus is not. $\endgroup$
    – David Loeffler
    Commented Jun 6, 2016 at 8:35
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    $\begingroup$ If a simply connected semisimple $k$-group $G$ has a Borel subgroup $B$ (defined over $k$), then any maximal torus $T\subset B\subset G$ is quasi-split in your sense. $\endgroup$
    – Mikhail Borovoi
    Commented Jun 6, 2016 at 8:35
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    $\begingroup$ It not true, that any maximal torus in a quasi-split group is quasi-split in your sense (consider a compact torus in the split group $\mathrm{SL}(2,\mathbb{R})$ ). $\endgroup$
    – Mikhail Borovoi
    Commented Jun 6, 2016 at 8:39
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    $\begingroup$ @David Loeffler: The determinant of an element of $U(2,2)$ is in $U(1)$. So, you get a fiber product over $U(1)$ instead of $\mathbf G_m$. The resulting torus is indeed quasi-split. $\endgroup$
    – Friedrich Knop
    Commented Jun 6, 2016 at 10:25
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    $\begingroup$ @DavidLoeffler: I can explain the proof in terms of coroots (this is the same as Knop writes). Let $\alpha_1^\vee,\alpha_2^\vee,\alpha_3^\vee$ be the simple coroots of $G=SU(2,2)$ with respect to $T$ and $B$. Since $G$ is simply connected, these simple coroots consitute a basis of the cocharacter group of $T$. Since $B$ is defined over $\mathbb{R}$, the complex conjugation permutes the simple coroots. Thus $T\simeq \mathbb{R}^*\times\mathbb{C}^*$, hence it is quasi-split. $\endgroup$
    – Mikhail Borovoi
    Commented Jun 6, 2016 at 10:31

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A torus $T$ is quasi-split if its character group is a permutation representation for the Galois group. So a counterexample to your question is: let $G$ be the quasi-split group $SO(n+1,n-1)$, $n\ge2$, $k=\mathbb R$. The weight lattice is $\mathbb Z^n$ with Galois action $$ (x_1,\ldots,x_{n-1},x_n)\mapsto(x_1,\ldots,x_{n-1},-x_n) $$ This is not a permutation module.

On the other hand, the Galois action on $T\subseteq B\subseteq G$ of a quasi-split group is induced by an action on the simple roots. These form a basis if $G$ is of adjoint type. For simply connected groups one can play the same game with fundamental representations. So:

If $G$ is quasi-split, and either simply connected or of adjoint type then the maximal $k$-torus sitting in a $k$-Borel is quasi-split.

I have the suspicion that the two notions of "quasi-splitness" developed independently.

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    $\begingroup$ Perhaps the term "quasi-split" (or "quasi-trivial") for tori arose because according to Shapiro's Lemma (and Hilbert 90) $H^1(F,R_{L/F}\mathbf{G}_m) = H^1(L,\mathbf{G}_m) = 1$ -- i.e. "the first galois cohomology group of a quasi-split torus is the same (i.e. is trivial) as that for a split torus". $\endgroup$
    – George McNinch
    Commented Jun 6, 2016 at 13:35
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    $\begingroup$ This fact for adjoint and simply connected cases was known prior to Harder's paper; e.g., it is Prop. 3.13 Exp. XXIV of SGA3 (expressing the maximal torus in a Borel subgroup as the Weil restriction of ${\rm{GL}}_1$ over the finite etale cover given by the "scheme of Dynkin diagrams"). The setup there may seem forbiddingly general, but the method of proof is exactly the same permutation argument. This fact must have been known to Borel, Tits, Steinberg, et al. prior to SGA3, but at least SGA3 provides an earlier reference. The terminology "quasi-split" for such tori is bad; please avoid it. $\endgroup$
    – nfdc23
    Commented Jun 6, 2016 at 14:52
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    $\begingroup$ @nfdc23. What is a better term than "quasi-split" for tori? That is an honest question; I do not know what is a good term. Would "permutation torus" be a better term? $\endgroup$
    – Jason Starr
    Commented Jun 7, 2016 at 12:37
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    $\begingroup$ @JasonStarr: "Induced torus". $\endgroup$
    – Mikhail Borovoi
    Commented Jun 7, 2016 at 13:59
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    $\begingroup$ @JasonStarr: Borovoi gave the response I would have given. The motivation for this alternative terminology is that the geometric character lattice of the Weil restriction of a torus through a finite separable extension of fields is the induced representation (for absolute Galois groups) of the geometric character lattice of the given torus. For a hands-on proof of that (natural) identification, see the proof of Theorem 2.4(a) in Chapter II of Oesterle's 1984 Inventiones paper "Nombres de Tamagawa...". $\endgroup$
    – nfdc23
    Commented Jun 8, 2016 at 3:19

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