Hypergeometric $\mathcal D$-modules and exponential sums for reductive groups

Abstract

We define the hypergeometric exponential sum associated to a family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the hypergeometric exponential sum. Motivated by the definition of the hypergeometric sheaf, we introduce the hypergeometric $\mathcal D$-module, prove it is holonomic and estimate its rank. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we show how the hypergeometric $\mathcal D$-module controls the general behavior of the hypergeometric sheaf. We apply our results to the estimation of the hypergeometric exponential sum.