Explicit functorial correspondences for level zero representations of p-adic linear groups

Let F be a non-Archimedean local field and D a central F-division algebra of dimension n2, n⩾1. We consider first the irreducible smooth representations of D× trivial on 1-units, and second the indecomposable, n-dimensional, semisimple, Weil–Deligne representations of F which are trivial on wild inertia. The sets of equivalence classes of these two sorts of representations are in canonical (functorial) bijection via the composition of the Jacquet–Langlands correspondence and the Langlands correspondence. They are also in canonical bijection via explicit parametrizations in terms of tame admissible pairs. This paper gives the relation between these two bijections. It is based on analysis of the discrete series of the general linear group GLn(F) in terms of a classification by extended simple types.