An explicit geometric Langlands correspondence for the projective line minus four points

Abstract

This article deals with the tamely ramified geometric Langlands correspondence for GL_2 on ${\mathbf{P}}_{{\mathbf{F}}_q}^1$, where $q$ is a prime power, with tame ramification at four distinct points $D=\{\infty ,0,1,t\}\subset {\mathbf{P}}^1({\mathbf{F}}_q)$. We describe in an explicit way (1) the action of the Hecke operators on a basis of the cusp forms, which consists of $q$ elements; and (2) the correspondence that assigns to a pure irreducible rank 2 local system $E$ on ${\mathbf{P}}^1\setminus D$ with unipotent monodromy its Hecke eigensheaf on the moduli space of rank 2 parabolic vector bundles. We define a canonical embedding ${\mathbf{P}}^1$ into this module space and show with a new proof that the restriction of the eigensheaf to the degree 1 part of this moduli space is the intermediate extension of $E$.

34 pages

Cited by in corpus (1)

• Geometric Langlands for hypergeometric sheaves