Hypergeometric sheaves for classical groups via geometric Langlands (Daxin Xu)
Time：2023-06-26 Source：Font Size：[Large | Medium | Small] [Print]
In a previous paper, the first and third authors gave an explicit realization of the geometric Langlands correspondence for hypergeometric sheaves, considered as GL(n)-local systems. Certain hypergeometric local systems admit a symplectic or orthogonal structure, which can be viewed as G-local systems, for a classical group G. This article aims to realize the geometric Langlands correspondence for these 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 G-local systems E_G on 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 E_G with certain G-opers on G_m. Finally, we compare these G-opers with hypergeometric local systems.
Publication:
Transactions of the American Mathematical Society
DOI: 10.1090/tran/8848
