Representation theory of \(\mathrm{GL}(n)\) over a \(p\)-adic division algebra and unitarity in the Jacquet-Langlands correspondence
type: Item
instance of: scholarly article
title: Representation theory of \(\mathrm{GL}(n)\) over a \(p\)-adic division algebra and unitarity in the Jacquet-Langlands correspondence
zbMATH Open document ID: 1124.22005
author: Marko Tadić
published in: Pacific Journal of Mathematics
publication date: 2007-02-26T00:00:00Z
review text: Let \(A\) be a central division algebra of dimension \(d_A\) over the local non-archimedean field \(F\) of characteristic \(0\). The generalized Jacquet-Langlands correspondence connects irreducible smooth representations of the group \(\text{GL}(p,A)\) with certain representations of \(\text{GL}(pd_A,F)\). This correspondence can be extended in a natural way to mappings between the Grothendieck groups of such representations. In the present paper the author assumes the truth of the conjecture that unitary parabolic induction is irrreducible for the group \(\text{GL}(m,A)\). Under this assumption he can prove explicit formulas for the correspondence. In particular it is proved that in the correspondence between the Grothendieck groups of \(\text{GL}(pd_A,F)\) and \(\text{GL}(p,A)\) an irreducible unitary representation of \(\text{GL}(pd_A,F)\) is sent either to zero or to plus or minus an irreducible representation of \(\text{GL}(p,A)\). On the way to this result some results in the representation theory of the group \(\text{GL}(n)\) of a division algebra over a non-archimedean local field are developed.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E35
zbMATH DE Number: 5129576
zbMATH Keywords: Representation theory
zbMATH Keywords: division algebra
zbMATH Keywords: Jacquet-Langlands correspondence
reviewed by: Rainer Schulze-Pillot
MaRDI profile type: Publication
full work available at URL: pjm.2006.223.167
OpenAlex ID: W1991185513
DOI: 10.2140/PJM.2006.223.167
description: scientific article; zbMATH DE number 5129576