Hypergeometric sheaves for classical groups via geometric Langlands

TL;DR

This paper extends the geometric Langlands correspondence to hypergeometric sheaves associated with classical groups, constructing $ ext{G}$-local systems via automorphic data and Hitchin systems, and relating them to hypergeometric local systems.

Contribution

It introduces a new framework for realizing the geometric Langlands correspondence for classical groups using hypergeometric automorphic data and Hitchin systems, including the first explicit construction of such $ ext{G}$-local systems.

Findings

Construction of hypergeometric automorphic data for classical groups.

Proof of rigidity of these automorphic data under certain conditions.

Identification of $ ext{G}$-local systems with hypergeometric local systems.

Abstract

In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as $\textrm{GL}_n$-local systems. Certain hypergeometric local systems admit a symplectic or orthogonal structure, which can be viewed as $\check{G}$-local systems, for a classical group $\check{G}$. This article aims to realize the geometric Langlands correspondence for these $\check{G}$-local systems. We study this problem from two aspects. In the first approach, we define the hypergeometric automorphic data for a classical group $G$ in the framework of Yun, one of whose local components is a new class of euphotic representations in the sense of Jakob-Yun. We prove the rigidity of hypergeometric automorphic data under natural assumptions, which allows us to define $\check{G}$-local systems $\mathcal{E}_{\check{G}}$ on $\mathbb{G}_m$ as Hecke eigenvalues (in both $\ell$-adic and de Rham setting). In the second approach (which works only in the de Rham setting), we quantize an enhanced ramified Hitchin system, following Beilinson-Drinfeld and Zhu, and identify $\mathcal{E}_{\check{G}}$ with certain $\check{G}$-opers on $\mathbb{G}_m$. Finally, we compare these $\check{G}$-opers with hypergeometric local systems.