12 deleted 8 characters in body

This question is interesting as it forces one to look closely at rational points of $F$-stable tori in $G$.

If $T$ is such torus then $F$ acts the lattice of cocharacters $X_*(T)$ thus inducing an automorphism of the set of coroots. Let $\sigma$ be this automorphism and suppose it has order $m$ (this is all we need to know about it). Let $\epsilon$ be a generator of the multiplicative group of the field in $q^m$ elements. Then for every coroot $\gamma^\vee$ we have that $\gamma^\vee(\epsilon)^F=(\sigma\gamma^\vee)(\epsilon^q)$. Now set $H_\gamma:=\prod_{i=0}^{m-1}(\sigma^i\gamma^\vee)(\epsilon^{q^i})$. Then $H_\gamma^F=H_\gamma$ by construction. Let $\alpha_1,\ldots, \alpha_l$ be a basis of simple roots in $X(T)$ and $a_1,\ldots, a_l\in\mathbb{Z}$. Define $H(a_1,\ldots, a_l):=H_{\alpha_1}^{a_1}\cdots H_{\alpha_l}^{a_l}$, an $F$-stable element of $T$.

We want fo find $a_i$ such that $H(a_1,\ldots, a_l)$ is regular (subject to some condition on $q$ to be determined). Let $a=\sum_{i-1}^la_i\alpha_i^\vee$, an element of $X_*(T)$, and let $e_\gamma\in g_\gamma$, a root vector in $g=Lie(G)$. Set $\tau=\sigma^{-1}$. Then $(Ad\ H(a_1,\ldots, a_l))\cdot e_\gamma= \epsilon^{M(a)}e_\gamma$ where $M(a)=\langle\gamma,\sum_{i=0}^{m-1}q^i\sigma^i(a)\rangle= \sum_{i=0}^{m-1}q^i\langle\tau^i\gamma,a\rangle$. Now $\epsilon$ is a primitive root of $1$ of order $q^m-1$. Suppose we found $a$ such that $\langle\beta,a\rangle\ne 0$ and $-(q-1)\le -(q-1)< \langle\beta,a\rangle\le langle\beta,a\rangle< q-1$ for all roots $\beta$. Then it follows from the uniqueness of $q$-adic expansions that $-(q^m-1)\le M(a)\le -(q^m-1)< M(a)< q^m-1$ and $M(a)\ne 0$, so that $H(a_1,\ldots, a_l)$ is regular as wanted. The condition on $a$ will depend on $q$, of course, but NOT on the choice of $T$, and it is quite easy to make it explicit.

Perhaps I should also mention that in the notation of Prop. 3.6.6 in Carter's book the condition $q\ge|\alpha|+1$ for all $\alpha$ is not always tha same as $q-1>ht(\alpha)$ for all positive roots $\alpha$ (especially when there are roots of different lengths).

11 I've corrected some misprints and improved exposition.

This question is interesting as it forces one to look closely at rational points of $F$-stable tori in $G$.

If $T$ is such torus then $F$ acts the lattice of cocharacters $X_*(T)$ thus inducing an automorphism of the set of coroots. Let $\sigma$ be this automorphism and suppose it has order $m$ (this is all we need to know about it). Let $\epsilon$ be a generator of the multiplicative group of the field in $q^m$ elements. Then for every coroot $\gamma^\vee$ we have that $\gamma^\vee(\epsilon)^F=(\sigma\gamma^\vee)(\epsilon^q)$. Now set $H_\gamma:=\prod_{i=0}^{m-1}(\sigma^i\gamma^\vee)(\epsilon^{q^i})$. Then $H_\gamma^F=H_\gamma$ by construction. Let $\alpha_1,\ldots, \alpha_l$ be a basis of simple roots in $X(T)$ and $a_1,\ldots, a_l\in\mathbb{Z}$. Define $H(a_1,\ldots, a_l):=H_{\alpha_1}^{a_1}\cdots H_{\alpha_l}^{a_l}$, an $F$-stable element of $T$.

We want fo find $a_i$ such that $H(a_1,\ldots, a_l)$ is regular (subject to some condition on $q$ to be determined). Let $a=\sum_{i-1}^la_i\alpha_i^\vee$, an element of $X_*(T)$, and let $e_\gamma\in g_\gamma$, a root vector in $g=Lie(G)$. Set $\tau=\sigma^{-1}$. Then $(Ad\ H(a_1,\ldots, a_l))\cdot e_\gamma= \epsilon^{M(a)}e_\gamma$ where $M(a)=\langle\gamma,\sum_{i=0}^{m-1}q^i\sigma^i(a)\rangle= \sum_{i=0}^{m-1}q^i\langle\tau\gamma,a\rangle$. sum_{i=0}^{m-1}q^i\langle\tau^i\gamma,a\rangle$. Now$\epsilon$is a primitive root of$1$of order$q^m-1$. Suppose we found$a$such that$\langle\beta,a\rangle\ne 0$and$-(q-1)\le \langle\beta,a\rangle\le q-1$for all roots$\beta$. Then it follows from the uniqueness of$q$-adic expansions that$-(q^m-1)\le M(a)\le q^m-1$and$M(a)\ne 0$, so that$H(a_1,\ldots, a_l)$is regular as wanted. The condition on$a$will depend on$q$, of course, but NOT on the choice of$T$, and it is quite easy to make it explicit. Perhaps I should also mention that in the notation of Prop. 3.6.6 in Carter's book the condition$q\ge|\alpha|+1$for all$\alpha$is not always tha same as$q-1>ht(\alpha)$for all positive roots$\alpha$(especially when there are roots of different lengths). 10 deleted 26 characters in body This question is interesting as it forces one to look closely at rational points of$F$-stable tori in$G$. If$T$is such torus then$F$acts the lattice of cocharacters$X_*(T)$thus inducing an automorphism of the set of coroots. Let$\sigma$be this automorphism and suppose it has order$m$(this is all we need to know about it). Let$\epsilon$be a generator of the multiplicative group of the field in$q^m$elements. Then , in Steinberg's notation, for every coroot$\gamma^\vee$we have that$\gamma^\vee(\epsilon)^F=(\sigma\gamma^\vee)(\epsilon^q)$. Now set$H_\gamma:=\prod_{i=0}^{m-1}(\sigma^i\gamma^\vee)(\epsilon^{q^i})$. Then$H_\gamma^F=H_\gamma$by construction. Let$\alpha_1,\ldots, \alpha_l$be a basis of simple roots in$X(T)$and$a_1,\ldots, a_l\in\mathbb{Z}$. Define$H(a_1,\ldots, a_l):=H_{\alpha_1}^{a_1}\cdots H_{\alpha_l}^{a_l}$, an$F$-stable element of$T$. We want fo find$a_i$such that$H(a_1,\ldots, a_l)$is regular (subject to some condition on$q$to be determined). Let$a=\sum_{i-1}^la_i\alpha_i^\vee$, an element of$X_*(T)$, and let$e_\gamma\in g_\gamma$, a root vector in$g=Lie(G)$. Set$\tau=\sigma^{-1}$. Then$(Ad\ H(a_1,\ldots, a_l))\cdot e_\gamma= \epsilon^{M(a)}e_\gamma$where$M(a)=\langle\gamma,\sum_{i=0}^{m-1}q^i\sigma^i(a)\rangle= \sum_{i=0}^{m-1}q^i\langle\tau\gamma,a\rangle$. Now$\epsilon$is a primitive root of$1$of order$q^m-1$. Suppose we found$a$such that$\langle\beta,a\rangle\ne 0$and$-(q-1)\le \langle\beta,a\rangle\le q-1$for all roots$\beta$. Then it follows from the uniqueness of$q$-adic expansions that$-(q^m-1)\le M(a)\le q^m-1$and$M(a)\ne 0$, so that$H(a_1,\ldots, a_l)$is regular as wanted. The condition on$a$will depend on$q$, of course, but NOT on the choice of$T$, and it is quite easy to make it explicit. Perhaps I should also mention that in the notation of Prop. 3.6.6 in Carter's book the condition$q\ge|\alpha|+1$for all$\alpha$is not always tha same as$q-1>ht(\alpha)$for all positive roots$\alpha\$ (especially when there are roots of different lengths).