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 

　　Author: 

　　Masoud Kamgarpour 

　　School of Mathematics and Physics, The University of Queensland.  

　　Daxin Xu 

　　Morningside Center of Mathematics and Hua Loo-Keng Key Laboratory of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 

　　School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China. 

　　E-mail address: daxin.xu@amss.ac.cn 

　　Lingfei Yi, School of Mathematics, University of Minnesota, Twin cities, Minneapolis, MN 55455