I want to also thank Zhiwei Yun and Konstantin Jakob for discussing and sharing their work on rigid automorphic data with me.Last, I want to thank my parents for their unconditional love, support, and encouragement.
Generalised hypergeometric sheaves are rigid local systems on the punctured projective line with remarkable properties. Their study originated in the seminal work of Riemann on the Euler-Gauss hypergeometric function and has blossomed into an active field with connections to many areas of mathematics. In this paper, we construct the Hecke eigensheaves whose eigenvalues are the irreducible hypergeometric local systems, thus confirming a central conjecture of the geometric Langlands program for hypergeometrics. The key new concept is the notion of hypergeometric automorphic data. We prove that this automorphic data is generically rigid (in the sense of Zhiwei Yun) and identify the resulting Hecke eigenvalue with hypergeometric sheaves. The definition of hypergeometric automorphic data in the tame case involves the mirabolic subgroup, while in the wild case, semistable (but not necessarily stable) vectors coming from principal gradings intervene.
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.
We construct a class of ℓ$\ell$‐adic local systems on double-struckA1$\mathbb {A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y′′(z)=zy(z)$y^{\prime \prime }(z)=zy(z)$. We employ the geometric Langlands correspondence to construct the sought‐after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô, and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GLn$\mathrm{GL}_n$, we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behavior of the local systems at ∞$\infty$. These conjectures, in particular, imply cohomological rigidity of Airy sheaves.
We construct a class of ℓ-adic local systems on A 1 that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y ′′ (z) = zy(z). We employ the geometric Langlands correspondence to construct the sought-after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GLn, we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behaviour of the local systems at ∞. These conjectures in particular imply cohomological rigidity of Airy sheaves.
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