Hypergeometric sheaves for classical groups via geometric Langlands

Abstract:

In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as $\mathrm {GL}_n$-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 settings). In the second approach (which works only in the de Rham setting), we quantize a 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.