The branching problem for generalized Verma modules, with application to the pair \((so(7), \operatorname{Lie}G_{2})\)
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 \(\iota(\bar{\mathfrak{g}})\), where \(\bar{\mathfrak{p}}\) is a parabolic subalgebra of \(\bar{\mathfrak{g}}\). If it is the case, then they also study a construction of certain singular vectors in \(M_\lambda(\mathfrak{g}, \mathfrak{p})\). In the last section they demonstrate their results by the example of \(\iota: \operatorname{Lie}(G_2) \hookrightarrow so(7)\).
scientific article; zbMATH DE number 6327975
scientific article