The branching problem for generalized Verma modules, with application to the pair \((so(7), \operatorname{Lie}G_{2})\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1304.22017
DOI: 10.1142/S0219498814500340
author: Todor Milev
author: Petr Somberg
publication date: 2014-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1209.3970
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 6327975
zbMATH Keywords: generalized Verma modules
zbMATH Keywords: branching problems
zbMATH Keywords: character formulas
zbMATH Keywords: singular vectors
zbMATH Keywords: non-symmetric pairs
zbMATH Keywords: \((so(7), \operatorname{Lie}G_{2})\)
Wikidata QID: Q115245666
MaRDI profile type: Publication
arXiv ID: 1209.3970
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published in: Journal of Algebra and its Applications
title: The branching problem for generalized Verma modules, with application to the pair \((so(7), \operatorname{Lie}G_{2})\)
review text: Let \(\mathfrak{g}\) be a complex simple Lie algebra and \(\mathfrak{p}\) a parabolic subalgebra of \(\mathfrak{g}\). Let \(M_\lambda(\mathfrak{g}, \mathfrak{p})\) denote the generalized Verma module for \(\mathfrak{g}\) induced from \(\mathfrak{p}\) with highest weight \(\lambda\). The branching problem for \(M_\lambda(\mathfrak{g}, \mathfrak{p})\) with respect to an embedding \(\iota:\bar{\mathfrak{g}} \hookrightarrow \mathfrak{g}\) of a reductive Lie algebra \(\bar{\mathfrak{g}}\) into \(\mathfrak{g}\) has received a lot of attention in representation theory and also in parabolic geometry. Here the branching problem for \(M_\lambda(\mathfrak{g}, \mathfrak{p})\) with respect to an embedding \(\iota:\bar{\mathfrak{g}} \hookrightarrow \mathfrak{g}\) means to study how the \(\mathfrak{g}\)-module \(M_\lambda(\mathfrak{g}, \mathfrak{p})\) behaves or decomposes as an \(\iota(\bar{\mathfrak{g}})\)-module. Such a study is carefully investigated especially when \(\iota(\bar{\mathfrak{g}})\) is strongly compatible (see, for example, \textit{T. Kobayashi} [Transform. Groups 17, No. 2, 523--546 (2012; Zbl 1257.22014)] and [\textit{T. Kobayashi} et al., ``Branching laws for Verma modules and applications in parabolic geometry. I,'' preprint (2013), \url{arXiv:1305.6040}]).NEWLINENEWLINEThe paper under review concerns a branching problem for \(M_\lambda(\mathfrak{g}, \mathfrak{p})\) not only for strongly compatible \(\iota(\bar{\mathfrak{g}})\) but also what the authors call weakly compatible \(\iota(\bar{\mathfrak{g}})\). If \(M_\lambda(\mathfrak{g}, \mathfrak{p})|_{\iota(\bar{\mathfrak{g}})}\) denotes \(M_\lambda(\mathfrak{g}, \mathfrak{p})\) as an \(\iota(\bar{\mathfrak{g}})\)-module, then, for such \(\iota(\bar{\mathfrak{g}})\), the authors study when \(M_\lambda(\mathfrak{g}, \mathfrak{p})|_{\iota(\bar{\mathfrak{g}})}\) decomposes into a direct sum of generalized Verma modules \(M_\mu(\iota(\bar{\mathfrak{g}}),\iota(\bar{\mathfrak{p}}))\) for \(\iota(\bar{\mathfrak{g}})\)…
reviewed by: Toshihisa Kubo
description: scientific article; zbMATH DE number 6327975