  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.
