Semisimple characters for inner froms I: GL_n(D)

TL;DR

This paper develops a theory of semisimple characters for inner forms of GL_n over non-archimedean fields, linking their classification to the Local Langlands correspondence and wild inertia behavior.

Contribution

It introduces semisimple characters for inner forms of GL_n, proves their intertwining properties, and classifies their classes via endo-parameters, advancing understanding of local representation theory.

Findings

Proved intertwining formula for semisimple characters

Established conjugacy-like theorem for intertwining

Classified intertwining classes using endo-parameters

Abstract

The article is about the representation theory of an inner form~$G$ of a general linear group over a non-archimedean local field. We introduce semisimple characters for~$G$ whose intertwining classes describe conjecturally via Local Langlands correspondence the behavior on wild inertia. These characters also play a potential role to understand the classification of irreducible smooth representations of inner forms of classical groups. We prove the intertwining formula for semisimple characters and an intertwining implies conjugacy like theorem. Further we show that endo-parameters for~$G$, i.e. invariants consisting of simple endo-classes and a numerical part, classify the intertwining classes of semisimple characters for~$G$. They should be the counter part for restrictions of Langlands-parameters to wild inertia under Local Langlands correspondence.