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Keywords:
generalized Verma modules; conformal geometry in dimension $5$; exceptional Lie algebra ${\mathrm{Lie~}G_2}$; F-method; branching problem
generalized Verma modules; conformal geometry in dimension $5$; exceptional Lie algebra ${\mathrm{Lie~}G_2}$; F-method; branching problem
Summary:
The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in [15]. In the present article, we employ the recently developed F-method, [10], [11] to the couple of non-compatible Lie algebras $\mathrm{Lie~}G_2\stackrel{i}{\hookrightarrow }{\operatorname{so}(7)}$, and generalized conformal ${\operatorname{so}(7)}$-Verma modules of scalar type. As a result, we classify the $i({\mathrm{Lie~}G_2}) \cap {\mathfrak{p}}$-singular vectors for this class of $\operatorname{so}(7)$-modules.
References:
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