Transfer of Representations and Orbital Integrals for Inner Forms of GLn

Abstract

AbstractWe characterize the Local Langlands Correspondence (LLC) for inner forms of GLn via the Jacquet–Langlands Correspondence (JLC) and compatibility with the Langlands Classification. We show that LLC satisfies a natural compatibility with parabolic induction and characterize LLC for inner forms as a unique family of bijections Π(GLr(D)) → Φ(GLr(D)) for each r, (for a fixed D), satisfying certain properties. We construct a surjective map of Bernstein centers ℨ(GLn(F)) → ℨ(GLr(D)) and show this produces pairs of matching distributions in the sense of Haines. Finally, we construct explicit Iwahori-biinvariant matching functions for unit elements in the parahoric Hecke algebras of GLr(D), and thereby produce many explicit pairs of matching functions.