Explicit Construction of Local Langlangds Correspondence of GL(2,F) Using Theta Correspondence

TL;DR

This paper explicitly constructs the local Langlands correspondence for GL(2,F) using theta correspondence and provides new proofs for its key properties, including independence from the additive character.

Contribution

It offers an explicit construction of the Langlands correspondence for GL(2,F) via theta correspondence and proves several of its fundamental properties.

Findings

Explicit construction of the correspondence

Proof of independence from additive character

New proofs of key properties

Abstract

The Langlands correspondence of GL(2,F) over a non-Archimedean local field F of characteristic 0 has been well studied. The construction uses the theta correspondence. In this paper, we are going to describe explicitly how this construction is done. We will then provide new proofs to several properties of the correspondence. Including that the correspondence does not depend on the additive character introduced by the Weil representation.