The geometric correspondence in some special cases

TL;DR

This paper explores specific cases of the geometric Langlands correspondence over complex curves, extending known results beyond the traditional finite field setting.

Contribution

It provides new proofs and extends the geometric Langlands correspondence to certain special cases over complex fields.

Findings

Proves the correspondence in some special cases over complex curves.

Extends the geometric Langlands framework beyond finite fields.

Provides new insights into the complex geometric Langlands program.

Abstract

The geometric Langlands correspondence for function fields over finite fields has been proved by Frenkel, Gaitsgory, Vilonen. The aim of this article is to write translation for curves over the complex field and prove the correspondence in some special cases and some new cases, outside the frame of the usual geometric Langlands correspondence.