Title: 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.