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I have had a lot of questions lately about root systems and rationality questions. Several people have referred me to the article Groupes Reductifs by Borel and Tits, which I am slowly reading. In the meantime, I had a question about relative roots for which I hope there is a simple explanation.

$G$ is a connected reductive group defined over a field $F$, $T$ is a maximal torus of $G$, $A_0$ the maximal $F$-split subtorus of $T$, $X_0$ the kernel of the restriction map $X = X(T) \rightarrow X(A_0)$, and $\Phi \subseteq X$ the set of roots of $T$ in $G$. Let $\mathfrak g$ be the Lie algebra of $G$.

Let $\Phi_0 = \Phi \cap X_0$, and let $\overline{\Phi}$ be the image of $\Phi - \Phi_0$ in $X/X_0 \cong X(A_0)$. Then $\overline{\Phi}$ is called the set of restricted roots of $T$ in $G$.

On the other hand, I have seen the restricted roots defined to be $\Phi(A_0,G)$. Here $\alpha \in X(A_0)$ lies in $\Phi(A_0,G)$ if and only if there exists a nonzero $X \in \mathfrak g$ (or maybe it is required to be in $\mathfrak g(F)$, I'm not sure) such that $\textrm{Ad } t X = \alpha(t)X$ for all $t \in A_0$.

It is clear that $$\overline{\Phi} \subseteq \Phi(A_0,G)$$ On the other hand, the converse inclusion seems to be asserting the following: for each $\alpha \in \Phi(A_0,G)$, and any $\chi \in X$ whose restriction to $A_0$ is $\alpha$, there exists an element $\chi_0 \in X_0$ such that $\chi + \chi_0 \in \Phi$. Should this be the case?

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  • $\begingroup$ Though your notation is somewhat different from that used by Borel and Tits in their various papers, it may help to look at the detailed $\S21$ in Borel's book Linear Algebraic Groups (Springer GTM 26, 1991) as well as the older Tits survey at the 1965 AMS Summer Institute in Boulder (AMS PSPM 9, 1967). Both sources include useful worked examples, some over local fields. What you ask about is seen to be true by following their definitions closely, though it's a complicated story and spills over into their follow-up IHES paper in 1972. $\endgroup$ – Jim Humphreys Nov 22 '16 at 23:29
  • $\begingroup$ Thank you, Borel's book has been very helpful. I seem to have worked out the answer to my question in an elementary way. $\endgroup$ – D_S Jan 11 '17 at 4:28
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As far as I can tell, there isn't anything fancy you need to prove the equivalence of these definitions beyond basic facts about rational representations of tori. Let $T$ be a torus, $S$ a closed subgroup, and $\pi: T \rightarrow \textrm{GL}(V)$ a rational representation. Then $V$ breaks up into a direct sum

$$V = \bigoplus\limits_{\chi \in X(T)} V_{\chi}$$

where $V_{\chi} = \{ v \in V: \pi(t)v = \chi(t)v \textrm{ for all } t \in T\}$. The restriction of $\pi$ to $S$ also breaks up $V$ into a direct sum

$$V = \bigoplus\limits_{\alpha \in X(S)} V_{\alpha}$$

where $V_{\alpha} = \{ v \in V: \pi(s)v = \alpha(s)v \textrm{ for all } s \in S\}$.

Lemma: for $\alpha \in X(S)$, we have

$$V_{\alpha} = \bigoplus\limits_{\chi \in X(T), \chi|S = \alpha} V_{\chi}$$

Proof: It is clear that the right hand side is contained in the left. Taking the direct sum over all $\alpha \in X(S)$, both sides are equal to $V$, hence each inclusion must be an equality. $\blacksquare$

The characters $\chi$ of $T$ for which $V_{\chi} \neq 0$ are called the weights of $T$ for $\pi$. The lemma implies that a character $\alpha$ of $S$ is a weight if and only if it is the restriction of a weight of $T$.

Now let $G$ be a connected, reductive group over $F$. Let $S$ be an $F$-split subtorus of a torus $T$ which is defined over $F$.

Proposition: Let $\alpha \in X(S)$. Then the following are equivalent:

1 . There exists a nonzero element $X \in \mathfrak g$, and a character $\chi \in X(T)$, such that $\chi|S = \alpha$, and $\textrm{Ad }t X = \chi(t)X$ for all $t \in T$.

2 . There exists a nonzero $X \in \mathfrak g$ such that $\textrm{Ad } s X = \alpha(s) X$ for all $s \in S$.

3 . There exists a nonzero $X \in \mathfrak g(F)$ such that $\textrm{Ad } s X = \alpha(s)X$ for all $s \in S$.

(1) $\Rightarrow$ (2) $\Rightarrow$ (3) is clear. (3) $\Rightarrow$ (2) comes from the rationality statement in 3.2.12(ii) (Springer, Linear Algebraic Groups), and the fact that the morphism $\textrm{Ad}:G \rightarrow \textrm{GL}(\mathfrak g)$ is defined over $F$. (2) $\Rightarrow$ (1) follows from the lemma. $\blacksquare$.

In particular, when $T$ is maximally split, and $S = A_0$, the maximal $F$-split subtorus of $T$, this gives three equivalent definitions of restricted roots.

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