Appears in collection : Reductive groups and automorphic forms. Dedicated to the French school of automorphic forms and in memory of Roger Godement.

Consider the local Jacquet-Langlands correspondence between the discrete series representations of a general linear group H over some non-Archimedean locally compact field F and those of an inner form G of H. Thanks to the description of the cuspidal representations of such groups by Bushnell-Kutzko's theory of types, one may ask for an explicit description of the local Jacquet -Langlands correspondence in terms of types. A major issue is then to describe how the correspondence behaves with respect to endoclasses. I will explain how this problem can be solved by introducing modular representation theory.