Transfer of Representations and Orbital Integrals for Inner Forms of $GL_n$

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 $\Pi(GL_r(D)) \to \Phi(GL_r(D))$ for each $r$, (for a fixed $D$) satisfying certain properties. We construct a surjective map of Bernstein centers $\mathfrak{Z}(GL_n(F))\to \mathfrak{Z}(GL_r(D))$ and show this produces pairs of matching distributions. 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.