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 G L n \mathrm {GL}_n -local systems. Certain hypergeometric local systems admit a symplectic or orthogonal structure, which can be viewed as G ˇ \check {G} -local systems, for a classical group G ˇ \check {G} . This article aims to realize the geometric Langlands correspondence for these G ˇ \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 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 G ˇ \check {G} -local systems E G ˇ \mathcal {E}_{\check {G}} on G m \mathbb {G}_m as Hecke eigenvalues (in both ℓ \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 E G ˇ \mathcal {E}_{\check {G}} with certain G ˇ \check {G} -opers on G m \mathbb {G}_m . Finally, we compare these G ˇ \check {G} -opers with hypergeometric local systems.