Quaternionic representations of exceptional Lie groups.
type: Item
instance of: scholarly article
title: Quaternionic representations of exceptional Lie groups.
zbMATH Open document ID: 1055.22010
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: The simply connected complex exceptional Lie groups \(F_4(\mathbf C),\, E_6(\mathbf C),\, E_7(\mathbf C),\, E_8(\mathbf C)\) contain, respectively, the connected quaternionic real forms \(F_{4,4},\, E_{6,4},\, E_{7,4},\, E_{8,4}\) with real root systems of type \(F_4\). \textit{B. H. Gross} and \textit{N. R. Wallach} [J. Reine Angew. Math. 481, 73--123 (1996; Zbl 0857.22012)] constructed three unitary representations \(\sigma\) of the double cover \(\tilde G\) of each group \(G = F_{4,4},\, E_{6,4}\mathbf Z_2,\, E_{7,4},\, E_{8,4}\). As in his previous paper [J. Funct. Anal. 172, No. 2, 377--403 (2000; Zbl 0953.22018)], the author considers the subgroups \(G' = \text{Spin}(5,4)\mathbf Z_2^2,\, \text{Spin}(5,4)\mathbf Z_2^3\subset F_{4,4},\; F_{4,4}\mathbf Z_2\subset E_{6,4}\mathbf Z_2,\; E_{6,4}U_1 \subset E_{7,4},\; E_{7,4}SU_2 \subset E_{8,4}\) and investigates the restrictions of \(\sigma\) onto their double covers \(\tilde G'\). In the present paper, the irreducible components of these restrictions are determined and certain dual pair correspondences associated with the dual pairs \(\tilde G'\) of representations of \(\tilde G\) are described.
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 2101114
zbMATH Keywords: exceptional Lie group
zbMATH Keywords: quaternionic real form
zbMATH Keywords: quaternionic discrete series
zbMATH Keywords: unitary representation
zbMATH Keywords: dual pair
author: Hung Yean Loke
MaRDI profile type: Publication
full work available at URL: pjm.2003.211.341
OpenAlex ID: W1974768603
Wikidata QID: Q115230657
DOI: 10.2140/PJM.2003.211.341
description: scientific article; zbMATH DE number 2101114