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12
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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
$-(q-1)\le -(q-1)< \langle\beta,a\rangle\le langle\beta,a\rangle< q-1$ 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).
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11
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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 $M(a)=\langle\gamma,\sum_{i=0}^{m-1}q^i\sigma^i(a)\rangle=
\sum_{i=0}^{m-1}q^i\langle\tau\gamma,a\rangle$.
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).
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10
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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 , 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).
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9
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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 characters cocharacters $X(T)$ 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 root coroot $\gamma$ \gamma^\vee$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. \gamma^\vee(\epsilon)^F=(\sigma\gamma^\vee)(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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).
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8
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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 characters $X(T)$ thus inducing an automorphism of the set of rootscoroots. 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 root $\gamma$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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).
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7
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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 characters $X(T)$ thus inducing an automorphism of the set of roots. 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 root $\gamma$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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 regulat 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).
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6
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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 characters $X(T)$ thus inducing an automorphism of $X(T)$ with preserves the set of roots. 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 root $\gamma$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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 regulat 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).
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5
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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 $F$ induces a Galois action of acts the lattice of characters $X(T)$ thus inducing an automorphism of $X(T)$ with preserves the set of roots. 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 root $\gamma$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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 regulat 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).
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4
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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 $F$ induces a Galois action of the lattice of characters $X(T)$ thus inducing an automorphism of $X(T)$ with preserves the set of roots. 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 root $\gamma$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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\gamma,a\rangle$.
Now $\epsilon$ is a primitive root of $1$ of order $q^m-1$. Suppose we found $a\in X_*(T)$ 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 $-(q^m-1)\le M(a)\le q^m-1$ and $M(a)\ne 0$, so that $H(a_1,\ldots, a_l)$ is regulat as wanted.
The condition on $a$ will depend on $q$, of course, but NOT of on the choice of $T$, and it is quite easy to make it explicit.
perhaps
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).
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3
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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 an $F$-stable such torus then $F$ induces a Galois action of the lattice of cocharacters characters $X_{*}(T)$ X(T)$ thus inducing an automorphism of $X_{*}(T)$ X(T)$ with preserves the set of corootsroots. Let $\sigma$ be this automorphism and suppose iy it has order $m$. 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 root $\gamma$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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 $a\in X_*(T)$ such that
$\langle\beta,a\rangle\ne 0$ and
$-(q-1)\le \langle\beta,a\rangle\le q-1$ for all roots $\beta$. Then $-(q^m-1)\le M(a)\le q^m-1$ and $M(a)\ne 0$, so that $H(a_1,\ldots, a_l)$ is regulat as wanted.
The condition on $a$ will depend on $q$, of course, but NOT of the choice of $T$, and it is easy to make it explicit.
perhaps I should 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).
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2
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If $T$ is an $F$-stable then $F$ induces a Galois action of the lattice of cocharacters $X_{*}(T)$ thus inducing an automorphism of $X_{*}(T)$ with preserves the set of coroots. Let $\sigma$ be this automorphism and suppose iy has order $m$. Let $\epsilon$ be a generator of the multiplicative group of the field in $q^m$ elements. Then, in Steinberg's notation, for every root $\gamma$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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 $a\in X_*(T)$ such that
$\langle\beta,a\rangle\ne 0$ and
$-(q-1)\le \langle\beta,a\rangle\le q-1$ for all roots $\beta$. Then $-(q^m-1)\le M(a)\le q^m-1$ and $M(a)\ne 0$, so that $H(a_1,\ldots, a_l)$ is regulat as wanted.
The condition on $a$ will depend on $q$, of course, but NOT of the choice of $T$, and it is easy to make it explicit.
perhaps I should also add mention that in the notation of Prop. 3.6.6 in Carter's book the condition $|q\ge\alpha|+1$ q\ge|\alpha|+1$ for all
$\alpha$ is not always tha same as $q>ht(\alpha)+1$q-1>ht(\alpha)$ for all positive roots $\alpha$ (especially when there are roots of different lengths).
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If $T$ is an $F$-stable then $F$ induces a Galois action of the lattice of cocharacters $X_{*}(T)$ thus inducing an automorphism of $X_{*}(T)$ with preserves the set of coroots. Let $\sigma$ be this automorphism and suppose iy has order $m$. Let $\epsilon$ be a generator of the multiplicative group of the field in $q^m$ elements. Then, in Steinberg's notation, for every root $\gamma$ we have that $h_\gamma(\epsilon)^F=h_{\sigma\gamma}(\epsilon^q)$. Now set
$H_\gamma:=\prod_{i=0}^{m-1}h_{\sigma^i\gamma}(\epsilon^{q^i})$. 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 $a\in X_*(T)$ such that
$\langle\beta,a\rangle\ne 0$ and
$-(q-1)\le \langle\beta,a\rangle\le q-1$ for all roots $\beta$. Then $-(q^m-1)\le M(a)\le q^m-1$ and $M(a)\ne 0$, so that $H(a_1,\ldots, a_l)$ is regulat as wanted.
The condition on $a$ will depend on $q$, of course, but NOT of the choice of $T$, and it is easy to make it explicit. I should also add that in Prop. 3.6.6 the condition $|q\ge\alpha|+1$ for all
$\alpha$ is not always tha same as $q>ht(\alpha)+1$
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