Breuil–Mézard conjectures for central division algebras

Abstract

We formulate an analogue of the Breuil–Mézard conjecture for the group of units of a central division algebra over a $p$-adic local field, and we prove that it follows from the conjecture for ${\mathrm{GL}<!--FUNCTION\; APPLICATION-->}_n$. To do so we construct a transfer of inertial types and Serre weights between the maximal compact subgroups of these two groups, in terms of Deligne–Lusztig theory, and we prove its compatibility with mod $p$ reduction, via the inertial Jacquet–Langlands correspondence and certain explicit character formulas. We also prove analogous statements for $\ell$-adic coefficients.