论文标题
Shintani Descent和Jordan分解的换向
Commutation of Shintani descent and Jordan decomposition
论文作者
论文摘要
令$ {\ mathbf g}^f $为有限的谎言类型组，其中$ {\ mathbf g} $是定义在$ {\ overline {\ mathbb f} _q} $的还原组，而$ f $是frobenius root。 Lusztig的Jordan分解参数在有理系列$ {\ Mathcal E}中参数不可减少字符$ {\ MATHCAL E}（C _ {{\ Mathbf G}^*}（s）^{f^*}，1）$。我们认为，Shintani扭曲保留了由$ {\ Mathcal e}（\ Mathcal e}（{\ Mathbff）联合产生的类函数的空间g}^f}，（s'）_ {{\ Mathbf g}^{*f^*}}）$其中$（s'）_ {{\ Mathbf g}^{*f^*}} $在$ {{\ Mathbf g}^use { $ s $;此外，通过线性将约旦分解扩展到这个空间，我们猜想有一种方法可以修复约旦分解，以使Shintani映射到Shintani twists twist the Shintani在Deshpande定义的断开群体上，该群体是在$ \ coprod_ {s's's's's'} s's'} coprod_的线性跨度上的。 e}（c _ {{\ mathbf g}^*}（s'）^{f^*}，1）$。我们展示了此猜想的一个非平凡情况，即$ {\ Mathbf G} $带有$ n $ prime的$ {\ mathbf g} $。
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.
