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

Published:2017-07-04

• Jonathan Cohen,
University of Maryland, Department of Mathematics, College Park, MD 20742-4015, USA
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## Abstract

We characterize the Local Langlands Correspondence (LLC) for inner forms of $\operatorname{GL}_n$ 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 $\Pi(\operatorname{GL}_r(D)) \to \Phi(\operatorname{GL}_r(D))$ for each $r$, (for a fixed $D$) satisfying certain properties. We construct a surjective map of Bernstein centers $\mathfrak{Z}(\operatorname{GL}_n(F))\to \mathfrak{Z}(\operatorname{GL}_r(D))$ 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 $\operatorname{GL}_r(D)$, and thereby produce many explicit pairs of matching functions.
 Keywords: Langlands correspondence, inner form
 MSC Classifications: 20G05 - Representation theory

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