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Let $G$ be a reductive group, $B_0$ a $F$-stable Borel subgroup and $T_0$ a fixed $F$-stable maximal torus contained in $B_0$. ($F$ = Frobenius morphism). Let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$.

For $w$ in the Weyl group associated with $T_0$, let $x \in G$ such that $x^{-1} F(x) = w$ and define $T_w = x T_0 x^{-1}$.

It is then stated in the article I'm reading that :

If the maximal torus $T_w$ is not contained in any proper $F$-stable parabolic subgroup of $G$, it is well known that this implies $l(w) = r\ ( mod\ 2 )$.

Does anyone has a reference for this ?

Edit (following Jim's advices) : The article I'm currently working on is "Representations of finite Chevalley groups" from Lusztig (1977) (CBMS Regional Conf. Series in Math.n°39). Even more specifically, I'm trying to understand many examples (3.10) that are not fully developped about unipotent representations. The previous quote is from the proof of the following claim :

Assume $q$ is greater than the Coxeter number of $G$. Let $\rho$ be an irreducible cuspidal $G^F$-submodule of $H^{i}_{c}(X_w)_{\mu}$ and let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$. Then all complex conjugates of $\mu$ have absolute value of the form $q^{\delta m /2}$ where $m$ is an integer congruent to $r$ modulo $2$.

3 edited body

Let $G$ be a reductive group, $B_0$ a Borel subgroup and $T_0$ a fixed $F$-stable maximal torus contained in $B_0$. ($F$ = Frobenius morphism). Let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$.

For $w$ in the Weyl group associated with $T_0$, let $x \in G$ such that $x^{-1} F(x) = w$ and define $T_w = x T_0 x^{-1}$.

It is then stated in the article I'm reading that :

If the maximal torus $T_w$ is not contained in any proper $F$-stable parabolic subgroup of $G$, it is well known that this implies $l(w) = r\ ( mod\ 2 )$.

Does anyone has a reference for this ?

Edit (following Jim's advices) : The article I'm currently working on is "Representations of finite Chevalley groups" from Lusztig (1976) 1977) (CBMS Regional Conf. Series in Math.n°39). Even more specifically, I'm trying to understand many examples (3.10) that are not fully developped about unipotent representations. The previous quote is from the proof of the following claim :

Assume $q$ is greater than the Coxeter number of $G$. Let $\rho$ be an irreducible cuspidal $G^F$-submodule of $H^{i}_{c}(X_w)_{\mu}$ and let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$. Then all complex conjugates of $\mu$ have absolute value of the form $q^{\delta m /2}$ where $m$ is an integer congruent to $r$ modulo $2$.

2 added 749 characters in body

Let $G$ be a reductive group, $B_0$ a Borel subgroup and $T_0$ a fixed $F$-stable maximal torus contained in $B_0$. ($F$ = Frobenius morphism). Let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$.

For $w$ in the Weyl group associated with $T_0$, let $x \in G$ such that $x^{-1} F(x) = w$ and define $T_w = x T_0 x^{-1}$.

It is then stated in the article I'm reading that :

If the maximal torus $T_w$ is not contained in any proper $F$-stable parabolic subgroup of $G$, it is well known that this implies $l(w) = r\ ( mod\ 2 )$.

Does anyone has a reference for this ?

Edit (following Jim's advices) : The article I'm currently working on is "Representations of finite Chevalley groups" from Lusztig (1976) (CBMS Regional Conf. Series in Math.n°39). Even more specifically, I'm trying to understand many examples (3.10) that are not fully developped about unipotent representations. The previous quote is from the proof of the following claim :

Assume $q$ is greater than the Coxeter number of $G$. Let $\rho$ be an irreducible cuspidal $G^F$-submodule of $H^{i}_{c}(X_w)_{\mu}$ and let $r$ be the semisimple $\mathbb{F}_q$-rank of $G$. Then all complex conjugates of $\mu$ have absolute value of the form $q^{\delta m /2}$ where $m$ is an integer congruent to $r$ modulo $2$.

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