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 $E$.
Categories: math.AG