On Ginzburg-Kaplan gamma factors and Bessel-Speh functions for finite general linear groups

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 of generic principal series representations and twisted matrix Kloosterman sums. Using this relation, we establish the multiplicativity identity of twisted matrix Kloosterman sums.