On representations of \({\mathfrak p}\)-adic split and non-split symplectic groups and their character relations
type: Item
instance of: scholarly article
title: On representations of \({\mathfrak p}\)-adic split and non-split symplectic groups and their character relations
zbMATH Open document ID: 0533.12015
author: Tatsuo Hina
author: Hiroshi Masumoto
published in: Proceedings of the Japan Academy. Series A
publication date: 1983-01-01T00:00:00Z
review text: Over a non-Archimedean local field F, the quaternionic unitary group with similitudes GUq(n) is \(\bar F\)-isomorphic to GSp(n), the symplectic group. For \(n=1\), \textit{H. Jacquet} and \textit{R. P. Langlands} [Automorphic forms on GL(2), Lect. Notes Math. 114 (1970; Zbl 0236.12010)] established a correspondence between certain of the irreducible admissible representations of GSp(1,F)\(\cong GL(2,F)\) and those of \(GUq(1,F)\cong D^{\times}\), where D is a division quaternion algebra over F. This note describes a similar correspondence, in terms of characters, for certain series of representations of GSp(2,F) and GUq(2,F), assuming the residue characteristic is odd. Both groups contain Cartan subgroups isomorphic to the multiplicative groups of quadratic extensions of F. These tori are contained in parabolic subgroups whose Levi components are essentially GL(2,F) or \(D^{\times}\), respectively. If one takes representations of these parabolic subgroups which are related by the Jacquet-Langlands correspondence and induces to GSp(2) and GUq(2) respectively, then it is reasonable to expect a relationship between the induced representations. The authors show their characters are each other's negatives, by calculating the characters explicitly.
Mathematics Subject Classification ID: 11S37
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 11S45
zbMATH DE Number: 3845711
zbMATH Keywords: division quaternion algebra over non-Archimedean local field
zbMATH Keywords: symplectic group with similitudes
zbMATH Keywords: relationship between induced representations
zbMATH Keywords: Jacquet-Langlands correspondence
reviewed by: Joe Repka
MaRDI profile type: Publication
full work available at URL: pjaa.59.195
OpenAlex ID: W2068944835
cites work: Q3898575
cites work: On certain arithmetical Dirichlet series
cites work: Automorphic forms on GL (2)
DOI: 10.3792/PJAA.59.195
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description: scientific article; zbMATH DE number 3845711