  In the present article, we combine some techniques in the harmonic analysis
together with the geometric approach given by modules over sheaves of rings of
twisted differential operators ($\mathcal{D}$-modules), and reformulate the
composition series and branching problems for objects in the
Bernstein-Gelfand-Gelfand parabolic category $\mathcal{O}^\mathfrak{p}$
geometrically realized on certain orbits in the generalized flag manifolds. The
general framework is then applied to the scalar generalized Verma modules
supported on the closed Schubert cell of the generalized flag manifold $G/P$
for $G={\rm SL}(n+2,\mathbb{C})$ and $P$ the Heisenberg parabolic subgroup, and
the algebraic analysis gives a complete classification of
$\mathfrak{g}'_r$-singular vectors for all
$\mathfrak{g}'_r=\mathfrak{sl}(n-r+2,\mathbb{C})\,\subset\,
\mathfrak{g}=\mathfrak{sl}(n+2,\mathbb{C})$, $n-r > 2$. A consequence of our
results is that we classify ${\rm SL}(n-r+2,\mathbb{C})$-covariant differential
operators acting on homogeneous line bundles over the complexification of the
odd dimensional CR-sphere $S^{2n+1}$ and valued in homogeneous vector bundles
over the complexification of the CR-subspheres $S^{2(n-r)+1}$.
