Semisimple characters for inner froms I: GL_n(D)

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.