Two definitions of restricted roots

Question

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?

Answer 1

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.