Transfer of Representations and Orbital Integrals for Inner Forms of GLn. (1st June 2018) - Cohen, Jonathan
Abstract: We characterize the Local Langlands Correspondence ( LLC ) for inner forms of GL_n 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 ( GL_r( D ) ) → Φ ( GL_r( D ) ) for each r, (for a fixed D ), satisfying certain properties. We construct a surjective map of Bernstein centers Z( GL_n( F ) ) → Z( GL_r( 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 GL_r( D ), and thereby produce many explicit pairs of matching functions.