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We discuss the branching problem for generalized Verma modules$M_\lambda(\mathfrak g, \mathfrak p)$ applied to couples of reductive Liealgebras $\bar{\mathfrak g}\hookrightarrow \mathfrak g$. The analysis is basedon projecting character formulas to quantify the branching, and on the actionof the center of $U(\bar{\mathfrak g})$ to explicitly construct singularvectors realizing part of the branching. We demonstrate the results on the pair$\mathrm{Lie}G_2\hookrightarrow{so(7)}$ for both strongly and weakly compatiblewith $i(\mathrm {Lie} G_2)$ parabolic subalgebras and a large class of inducingrepresentations.

The branching problem for generalized Verma modules, with application to the pair $(so(7),Lie G_2)$, extended version with tables