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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 $-(q-1)< \langle\beta,a\\rangle< q-1$ for all roots $\beta$. Then it follows from the uniqueness of $q$-adic expansions that $-(q^m-1)< M(a)< q^m-1$ 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).

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 $-(q-1)< \langle\beta,a\rangle< q-1$ for all roots $\beta$. Then it follows from the uniqueness of $q$-adic expansions that $-(q^m-1)< M(a)< q^m-1$ 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).

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 $-(q-1)\le \langle\beta,a\\rangle\le q-1$ 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).

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 $-(q-1)< \langle\beta,a\\rangle< q-1$ for all roots $\beta$. Then it follows from the uniqueness of $q$-adic expansions that $-(q^m-1)< M(a)< q^m-1$ 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).

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 $M(a)=\langle\gamma,\sum_{i=0}^{m-1}q^i\sigma^i(a)\rangle= \sum_{i=0}^{m-1}q^i\langle\tau\gamma,a\rangle$. 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).

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 $-(q-1)\le \langle\beta,a\\rangle\le q-1$ 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).