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

TL;DR

This paper characterizes the Local Langlands Correspondence for inner forms of $GL_n$ using the Jacquet-Langlands Correspondence, establishing compatibility with parabolic induction and constructing explicit matching functions.

Contribution

It provides a unique characterization of LLC for inner forms, constructs a surjective map of Bernstein centers, and explicitly produces matching functions for parahoric Hecke algebras.

Findings

Established a bijection between representations and Langlands parameters for inner forms.

Constructed a surjective map between Bernstein centers of different groups.

Produced explicit matching functions for Hecke algebras.

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.