Shintani下降的交换与若尔当分解
Commutation of Shintani descent and Jordan decomposition
摘要：
设${\mathbf G}^F$是有限Lie型群，其中${\mathbf G}$是定义在${\overline{\mathbb F}_q}$上的约化群，$F$是Frobenius根. Lusztig的若尔当分解是有理级数${\mathcal E}({{\mathbf G}^F},(s)_{{\mathbf G}^{*F^*}})$中不可约特征标的参数化，其中$s\in{{\mathbf G}^{*F^*}}$由级数${\mathcal E}(C_{{\mathbf G}^*}(s)^{F^*},1)$表示，我们猜想Shintani扭曲保持了由${\mathcal E}({{\mathbf G}^F},(s')_{{\mathbf G}^{*F^*}})$并生成的类函数空间，其中$(s')_{{\mathbf G}^{*F^*}}$运行在与$s$几何共轭的${{\mathbf G}^{*F^*}}$的半单类上;进一步地，将若尔当分解线性地推广到这个空间，我们猜想有一种方法可以修正若尔当分解，使得它将Shintani扭曲映射到Deshpande定义的不连通群上的Shintani扭曲，其作用于$\coprod_{s'}{\mathcal E}(C_{{\mathbf G}^*}(s')^{F^*},1)$的线性跨度。我们展示了一个非平凡的情况下，这个猜想，${\mathbf G}$的情况下，$A_{n-1}$与$n$素数的类型。
英文摘要：
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${\mathcal E}({{\mathbf G}^F},(s)_{{\mathbf G}^{*F^*}})$ where $s\in{{\mathbf G}^{*F^*}}$ by the series ${\mathcal E}(C_{{\mathbf G}^*}(s)^{F^*},1)$.We conjecture that the Shintani twisting preserves the space of class functions generated by the union of the ${\mathcal E}({{\mathbf G}^F},(s')_{{\mathbf G}^{*F^*}})$ where$(s')_{{\mathbf G}^{*F^*}}$ runs over the semi-simple classes of ${{\mathbf G}^{*F^*}}$ 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 $\coprod_{s'}{\mathcal E}(C_{{\mathbf G}^*}(s')^{F^*},1)$. We show a non-trivial case of this conjecture, the case where ${\mathbf G}$ is of type $A_{n-1}$with $n$ prime.
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