Jacquet modules of principal series generated by the trivial \(K\)-type
type: Item
instance of: scholarly article
title: Jacquet modules of principal series generated by the trivial \(K\)-type
zbMATH Open document ID: 1168.22013
author: Noriyuki Abe
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2009-04-14T00:00:00Z
review text: Let \(G\) be a real semisimple Lie group and \(X\) be an irreducible admissible representation of \(G\). In 1980, \textit{W. Casselman} [Proc. int. Congr. Math., Helsinki 1978, Vol.~2, 557--563 (1980; Zbl 0425.22019)] introduced the Jacquet module \(J(X)\) of \(X\). The structure of the Jacquet module \(J(X)\) turns out to be quite complicated and can only be determined in very special cases. If \(G\) has real rank 1, then the structure of \(J(X)\) is known in detail [see \textit{D. H. Collingwood}, Representations of rank one Lie groups. Research Notes in Mathematics, 137. Boston - London - Melbourne (1985; Zbl 0647.22007)]. In the paper under review the author determines generators of the Jacquet module of a principal series representation by the trivial \(K\)-type.
Mathematics Subject Classification ID: 22E47
zbMATH DE Number: 5543226
zbMATH Keywords: Jacquet module
zbMATH Keywords: Harish-Chandra module
zbMATH Keywords: semisimple Lie group
zbMATH Keywords: principal series representation
zbMATH Keywords: generator
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1231263783
OpenAlex ID: W1982299766
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DOI: 10.2977/PRIMS/1231263783
description: scientific article; zbMATH DE number 5543226