Hypergeometric $\ell$-adic sheaves for reductive groups

TL;DR

This work constructs and analyzes hypergeometric $\ell$-adic sheaves attached to representations of a reductive group, encoding hypergeometric exponential sums as a family over a vector of endomorphisms. Central tools are Deligne-Fourier and Wang-Fourier transforms, which express hypergeometric sheaves as Fourier transforms of equivariant pushforwards and relate $\ell$-adic objects to $\mathcal{D}$-modules in characteristic zero. A nondegeneracy condition tied to the Newton polytope ensures that the hypergeometric sheaves are perverse with controlled ranks, yielding explicit bounds via Weyl-theoretic integrals and enabling Frobenius-based estimates of the sums. The results bridge the $\ell$-adic and $\mathcal{D}$-module formalisms, providing a framework to estimate hypergeometric exponential sums over reductive groups and to understand their geometric monodromy and dimensions. The paper also lays out a program to extend these bounds beyond the homogeneous setting through base-change arguments and regular holonomic D-modules.

Abstract

We define the hypergeometric exponential sum associated to a finite family of representations of a reductive group over a finite field. We introduce the hypergeometric $\ell$-adic sheaf to describe the behavior of the hypergeometric exponential sum. It is a perverse sheaf, and it is the counterpart in characteristic $p$ of the $A$-hypergeometric $\mathcal D$-module introduced by Kapranov. Using the theory of the Fourier transform for vector bundles over a general base developed by Wang, we are able to study the hypergeometric $\ell$-adic sheaf via the hypergeometric $\mathcal D$-module. We apply our results to the estimation of the hypergeometric exponential sum.

Theorems & Definitions (60)

• Proposition 4
• Corollary 5
• Theorem 7
• Proposition 1.2
• proof
• Definition 1.4
• Remark 1.5
• Definition 1.6
• Proposition 1.7
• proof