Transfer of Representations and Orbital Integrals for Inner Forms of GLn. (1st June 2018)

• Record Type: Journal Article
• Authors: Cohen, Jonathan
• Is Part Of: Canadian journal of mathematics. Volume 70:Number 3(2018)
• Page Start: 595
• Page End: 627
• Publication Date: 2018-06-01
• Subjects: 20G05; Langlands correspondence -- inner form; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2017-017-5
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: We characterize the Local Langlands Correspondence $\left( \text{LLC} \right)$ for inner forms of $\text{G}{{\text{L}}_{n}}$ via the Jacquet–Langlands Correspondence $\left( \text{JLC} \right)$ and compatibility with the Langlands Classification. We show that $\text{LLC}$ satisfies a natural compatibility with parabolic induction and characterize $\text{LLC}$ for inner forms as a unique family of bijections $\prod \left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)\, \to \, \Phi \left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)$ for each $r$, (for a fixed $D$ ), satisfying certain properties. We construct a surjective map of Bernstein centers $\mathfrak{Z}\left( \text{G}{{\text{L}}_{n}}\left( F \right) \right)\, \to \, \mathfrak{Z}\left( \text{G}{{\text{L}}_{r}}\left( D \right) \right)$ 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 $\text{G}{{\text{L}}_{r}}\left( D \right)$, and thereby produce many explicit pairs of matching functions.