On the local Langlands and Jacquet–Langlands correspondences

Abstract

Let F be a locally compact non Archimedean field, and D a division algebra with centre F and finite dimension d2 over F. Fix an integer r ≥ 1, and let G = GLn(F), G' = GLr(D), where n = rd. Smooth irreducible representations of G' are related to those of G via the Jacquet–Langlands correspondence, whereas the Langlands correspondence relates such representations of G to degree n representations of the absolute Galois group of F . We review some recent results on those correspondences, in particular on their explicit description.