The branching problem for generalized Verma modules, with application to the pair $(so(7),Lie{G}_2)$, extended version with tables

We discuss the branching problem for generalized Verma modules $M_{\lambda }(\mathfrak{g},\mathfrak{p})$ applied to couples of reductive Lie algebras $\bar{\mathfrak{g}}\hookrightarrow \mathfrak{g}$. The analysis is based on projecting character formulas to quantify the branching, and on the action of the center of $U(\bar{\mathfrak{g}})$ to explicitly construct singular vectors realizing part of the branching. We demonstrate the results on the pair $\mathrm{L}\mathrm{i}\mathrm{e}{G}_2\hookrightarrow so(7)$ for both strongly and weakly compatible with $i(\mathrm{L}\mathrm{i}\mathrm{e}{G}_2)$ parabolic subalgebras and a large class of inducing representations.