Let ${\mathbf{G}}^F$ be a finite group of Lie type, where $\mathbf{G}$ is a reductive group defined over ${\overline{\mathbb{F}}}_q$ and $F$ is a Frobenius root. Lusztig’s Jordan decomposition parametrises the irreducible characters in a rational series $\mathcal{E}({\mathbf{G}}^F,{\left(s\right)}_{{\mathbf{G}}^{\ast F^{\ast }}})$ where $s\in {\mathbf{G}}^{\ast F^{\ast }}$ by the series $\mathcal{E}(C_{{\mathbf{G}}^{\ast }}{\left(s\right)}^{F^{\ast }},1)$. We conjecture that the Shintani twisting preserves the space of class functions generated by the union of the $\mathcal{E}({\mathbf{G}}^F,{\left(s^{\prime }\right)}_{{\mathbf{G}}^{\ast F^{\ast }}})$ where ${\left(s^{\prime }\right)}_{{\mathbf{G}}^{\ast F^{\ast }}}$ runs over the semi-simple classes of ${\mathbf{G}}^{\ast F^{\ast }}$ geometrically conjugate to $s$; further, extending the Jordan decomposition by linearity to this space, we conjecture that there is a way to fix Jordan decomposition such that it maps the Shintani twisting to the Shintani twisting on disconnected groups defined by Deshpande, which acts on the linear span of ${\coprod }_{s^{\prime }}\mathcal{E}(C_{{\mathbf{G}}^{\ast }}{\left(s^{\prime }\right)}^{F^{\ast }},1)$. We show a non-trivial case of this conjecture, the case where $\mathbf{G}$ is of type $A_{n-1}$ with $n$ prime.

让 ${\mathbf{G}}^F$ 是一个 Lie 类型的有限群，其中 $\mathbf{G}$ 是定义在的还原组 ${\overline{\mathbb{F}}}_q$ 和 $F$是 Frobenius 根。Lusztig 的 Jordan 分解参数化了有理数列中的不可约字符$\mathcal{乙}({\mathbf{G}}^F,{\left(秒\right)}_{{\mathbf{G}}^{＊F^{＊}}})$ 在哪里 $秒\in {\mathbf{G}}^{＊F^{＊}}$ 按系列 $\mathcal{乙}(C_{{\mathbf{G}}^{＊}}{\left(秒\right)}^{F^{＊}},1)$. 我们推测 Shintani 扭曲保留了由$\mathcal{乙}({\mathbf{G}}^F,{\left({秒}^{\prime }\right)}_{{\mathbf{G}}^{＊F^{＊}}})$ 在哪里 ${\left({秒}^{\prime }\right)}_{{\mathbf{G}}^{＊F^{＊}}}$ 运行在半简单的类 ${\mathbf{G}}^{＊F^{＊}}$ 几何共轭 $秒$; 进一步，通过线性将 Jordan 分解扩展到这个空间，我们推测有一种方法可以修复 Jordan 分解，使得它将 Shintani 扭曲映射到 Deshpande 定义的不连接群上的 Shintani 扭曲，它作用于${\coprod }_{{秒}^{\prime }}\mathcal{乙}(C_{{\mathbf{G}}^{＊}}{\left({秒}^{\prime }\right)}^{F^{＊}},1)$. 我们展示了这个猜想的一个非平凡案例，其中$\mathbf{G}$ 是类型 ${\mathrm{一种}}_{n-1}$ 和 $n$ 主要的。