On a characteristic function of the tensor \(K\)-module of inner type noncompact real simple groups
type: Item
instance of: scholarly article
title: On a characteristic function of the tensor \(K\)-module of inner type noncompact real simple groups
zbMATH Open document ID: 1039.22008
author: Hisaichi Midorikawa
published in: Tokyo Journal of Mathematics
publication date: 2004-05-18T00:00:00Z
review text: Midorikawa considers a simply connected complex Lie group \(G_c\) together with some noncompact real version \(G\) of it. Let \(K\) be a maximal compact subgroup of \(G\) and \({k}\oplus{p}\) the related Cartan decomposition of its Lie algebra. Then the complexified version \({k}_c\oplus{p}_c\) decomposes the complex Lie algebra of \(G_c\). If \(V\) is a \(K\)-module, so is \(L={ p}_c\otimes V\). Let \(V\) be simple with heighest weight \(\mu\). Moreover, if \( b\) is a maximal abelian subalgebra of \( k\), one may consider the root system of \({ b}_c\) and obtain a decomposition \(L=\oplus L_\omega\) where \(\omega\) runs over all noncompact roots such that \(\mu+\omega\) is compact. Either \(L_\omega\) is a simple \(K\)-submodule or trivial, i.e.\ \(L_\omega=\{0\}\). The purpose of the paper is to determine when it is nontrivial by checking whether some rational function is nonzero. The tensor space \({ p}_c\otimes V\) has been used in the literature to classify the unitary representations of \(G\) when \(G\) is either \(SL(2,R)\), the de Sitter group, \(SO(2n,1)\) or \(SU(n,1)\).
Mathematics Subject Classification ID: 22E46
Mathematics Subject Classification ID: 22E60
zbMATH DE Number: 2064099
zbMATH Keywords: simple Lie groups
zbMATH Keywords: unitary representations
reviewed by: Gert Roepstorff
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1244208686
OpenAlex ID: W2035192605
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DOI: 10.3836/TJM/1244208686
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description: scientific article; zbMATH DE number 2064099