Branching problems and ${\mathfrak s}{\mathfrak l}(2,{\mathbb C})$-actions

We study certain ${\mathfrak s}{\mathfrak l}(2,{\mathbb C})$-actions associated to specific examples of branching of scalar generalized Verma modules for compatible pairs $({\mathfrak g},{\mathfrak p})$, $({\mathfrak g}',{\mathfrak p}')$ of Lie algebras and their parabolic subalgebras.