The local Langlands correspondence for inner forms of $SL_n$

TL;DR

This paper establishes the local Langlands correspondence for all inner forms of SL_n over non-archimedean local fields, extending known results and employing the method of close fields for positive characteristic cases.

Contribution

It provides a bijective correspondence between Langlands parameters and irreducible admissible representations for inner forms of SL_n, including archimedean and positive characteristic fields.

Findings

Established the correspondence for all inner forms of SL_n over non-archimedean fields.

Extended the correspondence to archimedean fields.

Demonstrated the compatibility of the close fields method with the Langlands correspondence.

Abstract

Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group $SL_n (F)$. It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for $SL_n (F)$ enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of $SL_n (F)$. An analogous result is shown in the archimedean case. To settle the case where F has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of $GL_n (F)$, when the fields are close enough compared to the depth of the representations.