Euphotic representations and rigid automorphic data

Rigidity in automorphic representations and local systems

We introduce the notion of rigidity for automorphic representations of groups over global function fields. We construct the Langlands parameters of rigid automorphic representations explicitly as local systems over open curves. We expect these local systems to be rigid. Examples of rigid automorphic representations from previous work are reviewed and more examples for $GL_{2}$ are discussed in details.

Geometric Langlands for hypergeometric sheaves

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 …

Epipelagic representations and rigid local systems

We construct automorphic representations for quasi-split groups $G$ over the function field $F=k(t)$ one of whose local components is an epipelagic representation in the sense of Reeder and Yu. We also construct the attached Galois representations under the Langlands correspondence. These Galois representations give new classes of conjecturally rigid, wildly ramified $^{L}G$-local systems over $\mathbb{P}^{1}-\{0,\infty\}$ that generalize the Kloosterman sheaves constructed earlier by Heinloth, Ngô and the author. We study the monodromy of these local systems and compute all examples when $G$ …

不规则θ联络与上同调表示的几何朗兰兹对应

Geometric Langlands for Irregular Theta Connections and Epipelagic Representations

本文在稳定分次为内分次且Kac坐标s₀为正的假设下，证明了不规则θ联络与对应刚性自守数据的Hecke特征层的几何朗兰兹对应，并给出了相关应用。

曲线上刚性 G-局部系统的动形实现与缓分歧几何 Langlands 对应

Motivic realization of rigid G-local systems on curves and tamely ramified geometric Langlands

本文证明了具有拟单值单值性和有限阶阿贝尔化的复不可约刚性 G-局部系统是动形的，推广了 Katz 的定理，并在 de Rham 几何 Langlands 设定下证明了 Hecke 特征层的存在性、谱分解及刚性与自守刚性的关系。

A rigid irregular connection on the projective line

In this paper we construct a connection on the trivial G-bundle on the projective line for any simple complex algebraic group G, which is regular outside of the points 0 and infinity, has a regular singularity at the point 0, with principal unipotent monodromy, and has an irregular singularity at the point infinity, with slope 1/h, the reciprocal of the Coxeter number of G. This connection, which admits the structure of an oper in the sense of Beilinson and Drinfeld, appears to be the characteristic 0 counterpart of a hypothetical family of l-adic representations, which should parametrize a sp…