Title: On Ginzburg-Kaplan gamma factors and Bessel-Speh functions for finite general linear groups Show Chinese title

Title: 关于Ginzburg-Kaplan伽马因子和有限一般线性群的贝塞尔-斯佩希函数

Abstract: We give a new construction of tensor product gamma factors for a pair of irreducible representations of $\operatorname{GL}_c\left(\mathbb{F}_q\right)$ and $\operatorname{GL}_k\left(\mathbb{F}_q\right)$. This construction is a finite field analog of a construction of doubling type due to Kaplan in the local field case and due to Ginzburg in the global case, and it only assumes that one of the representations in question is generic. We use this construction to establish a relation between special values of Bessel functions attached to Speh representations and exotic matrix Kloosterman sums. Using this relation, we establish various identities, including the multiplicativity identity of exotic matrix Kloosterman sums. Show Chinese abstract

Abstract: 我们给出了一对不可约表示的张量积伽马因子的新构造，这些表示属于$\operatorname{GL}_c\left(\mathbb{F}_q\right)$和$\operatorname{GL}_k\left(\mathbb{F}_q\right)$。 此构造是局部域情况下 Kaplan 所做的倍增型构造以及全球情况下 Ginzburg 所做的构造的有限域类比，并且它仅假设所涉及的表示之一是典型的。 我们使用此构造来建立与 Speh 表示相关的贝塞尔函数的特殊值和奇异矩阵 Kloosterman 和之间的关系。 利用这种关系，我们建立了各种恒等式，包括奇异矩阵 Kloosterman 和的乘法恒等式。