论文标题
关于有限一般线性组的通用表示的贝塞尔函数的值
On values of the Bessel function for generic representations of finite general linear groups
论文作者
论文摘要
我们找到了S. I. Gelfand的Bessel功能的递归表达式，用于$ \ operatatorName {gl} _n \ left（\ Mathbb {f} _q \ right）$的不可约的通用表示。我们表明，贝塞尔函数的特殊值可以实现为与异国koolosterman sums相关的$ l $函数的系数，也可以作为Katz Exotic Kloosterman Sheaves的外部力量的痕迹。作为应用程序，我们表明某些多项式具有特殊的贝塞尔功能作为其系数的值，它们的所有根都位于单位圆上。作为另一个应用程序，我们表明，不可减少通用表示的Shintani基础变化的贝塞尔函数的特殊值与通过迪克森多条件的代表性的贝塞尔函数的特殊值有关。
We find a recursive expression for the Bessel function of S. I. Gelfand for irreducible generic representations of $\operatorname{GL}_n\left(\mathbb{F}_q\right)$. We show that special values of the Bessel function can be realized as the coefficients of $L$-functions associated with exotic Kloosterman sums, and as traces of exterior powers of Katz's exotic Kloosterman sheaves. As an application, we show that certain polynomials, having special values of the Bessel function as their coefficients, have all of their roots lying on the unit circle. As another application, we show that special values of the Bessel function of the Shintani base change of an irreducible generic representation are related to special values of the Bessel function of the representation through Dickson polynomials.
