An Explicit Geometric Langlands Correspondence for the Projective Line Minus Four Points

Abstract

Abstract This article deals with the tamely ramified geometric Langlands correspondence for ${\operatorname{GL}}_2$ on $ {\textbf{P}}_{ {\textbf{F}}_q}^1$, where $q$ is a prime power, with tame ramification at four distinct points $D = \{\infty , 0,1, t\} \subset {\textbf{P}}^1( {\textbf{F}}_q)$. We describe in an explicit way (1) the action of the Hecke operators on a basis of the space of cusp forms (Theorem 7.3) and (2) the correspondence that assigns to a pure irreducible rank 2 local system $E$ on $ {\textbf{P}}^1 \setminus D$ with unipotent monodromy its Hecke eigensheaf ${\operatorname{Aut}}_E$ on the moduli space ${\operatorname{Bun}}_{2,D}$ of rank 2 parabolic vector bundles (Theorem 1.2). We define a canonical embedding $ {\textbf{P}}^1 \setminus D \hookrightarrow{\operatorname{Bun}}_{2,D}^{1}$ and show with a new proof that ${\operatorname{Aut}}_E|_{{\operatorname{Bun}}_{2,D}^{1}}$ is the intermediate extension of $E$.