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.