Let $G$ be a connected reductive group over a field $k$ (not necessarily algebraically closed). Let $S$ be a maximal split torus in $G$ with relative root system $\Phi = \Phi\_k(S,G)$. Let $\Phi^+$ be a choice of positive roots. A one-parameter subgroup $G\_m \rightarrow S$ acts on each root space as multiplication by $t^k$ for some $k \in \mathbb{Z}$.


Question: Must there exist a one-parameter subgroup $G\_m \rightarrow S$ that acts on each positive root space as multiplication by $t^k$ with $k \geq 1$ (the $k$ can depend on the positive root space, of course, but should always be a positive integer).


I apologize if this is phrased in a clumsy way -- I don't work in this area, but I need this for something else.



Action of split torus on positive root spaces
Yes: take the one-parameter subgroup that is the sum of positive coroots. It acts on each simple root space by $t \mapsto t^2$, so by a positive integer power of $t$ on each positive root space.


Incidentally, this is a particularly natural choice, but, even if you didn't know that it existed, you could construct an only slightly arbitrary choice yourself; simply take the basis $\{\varpi\_\alpha^\vee \mathrel: \alpha \in \Delta\}$ of $X\_\*(S) \otimes\_{\mathbb Z} \mathbb Q$ to the basis $\Delta$ of simple roots (usually called the fundamental coweights). Then $\rho^\vee = \sum \varpi\_\alpha^\vee$ pairs positively with every simple root, hence with every positive root; but *a priori* it lies only in $X\_\*(S) \otimes\_{\mathbb Z} \mathbb Q$, not in $X\_\*(S)$. If all you need is *some* cocharacter, then there is an easy fix; just take a suitable positive-integer multiple of it. (The lack of specificity about the multiplier there is what I mean by "only slightly arbitrary".)


If you care about which integer multiple you take, then it's only slightly more complicated. Sometimes $\rho^\vee$ itself is a cocharacter (for example, if $G$ is adjoint, such as $\rho^\vee : t \mapsto \operatorname{diag}(t, 1)$ for $G = \operatorname{PGL}\_2$), but not always (for example, for $G = \operatorname{SL}\_2$, we have that $2\rho^\vee : t \mapsto \operatorname{diag}(t, t^{-1})$ is not divisible in the character lattice); but, even when it isn't, certainly $2\rho^\vee$ *is* a cocharacter—namely, the sum of the positive coroots.


