On exotic matrix exponential sums and Bessel-Speh functions

In a previous work with Carmon, we defined Bessel--Speh functions.

These are matrix coefficients of irreducible Speh representations of $\mathrm{GL}_{kc}(\mathbb{F})$, where $\mathbb{F}$ is a finite field.

They arise from $(k,c)$ models, which are models that generalize the Whittaker model to Speh representations attached to irreducible generic representations.

These constructions are finite field analogs of objects arising naturally in the generalized doubling method over $p$-adic fields, a recently active area of the Langlands program.

In this article we study special values of Bessel--Speh functions which were used in our previous work with Carmon to define Ginzburg--Kaplan gamma factors.

Our main result computes the special values of interest explicitly in terms of new arithmetic objects we introduce, called exotic matrix Kloosterman sums, which generalize both Katz's exotic Kloosterman sums and twisted matrix Kloosterman sums.

We then show that exotic matrix Kloosterman sums can be expressed as products of modified Hall--Littlewood polynomials evaluated at roots of the characteristic polynomial of the Frobenius acting on Katz's exotic Kloosterman sheaf.

As an application of our results, we establish new identities for Bessel functions of irreducible generic representations.