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

Jon Cohen

paper

Transfer of Representations and Orbital Integrals for Inner Forms of $G L_{n}$

arXiv:1606.02141

Abstract

We characterize the Local Langlands Correspondence (LLC) for inner forms of $G L_{n}$ 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 $Π (G L_{r} (D)) \to Φ (G L_{r} (D))$ for each $r$ , (for a fixed $D$ ) satisfying certain properties. We construct a surjective map of Bernstein centers $Z (G L_{n} (F)) \to Z (G L_{r} (D))$ 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 $G L_{r} (D)$ , and thereby produce many explicit pairs of matching functions.

39 pages

Transfer of Representations and Orbital Integrals for Inner Forms of GLn​

Abstract

Transfer of Representations and Orbital Integrals for Inner Forms of $G L_{n}$