Breuil-Mézard conjectures for central division algebras

Abstract

We give a parametrization of the inertial classes of smooth representations of inner forms of GL(n) over a p-adic field, based on type-theoretic invariants. Then we give a complete description of the behaviour of this parametrization under the Jacquet–Langlands correspondence, proving a conjecture of Broussous, Sécherre and Stevens on preservation of endo-classes. As an application of this result, we construct a Jacquet–Langlands transfer of types and Serre weights for central division algebras, and use it to deduce a form of the Breuil–Mézard conjecture, for discrete series Galois deformation rings and types of central division algebras, from the conjectural statement for GL(n).