A regular normal parabolic geometry of type $G/P$ on a manifold $M$ gives rise to sequences $D_i$ of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ${\nabla }^{\omega }$ on the corresponding tractor bundle $V$, where $\omega$ is the normal Cartan connection. The first operator $D_0$ in the sequence is overdetermined and it is well known that ${\nabla }^{\omega }$ yields the prolongation of this operator in the homogeneous case $M=G/P$. Our first main result...

We discuss some aspects of the composition structure of twisted Verma modules for the Lie algebra $\mathrm{𝔰𝔩}(3,ℂ)$, including the explicit structure of singular vectors for both $\mathrm{𝔰𝔩}(3,ℂ)$ and one of its Lie subalgebras $\mathrm{𝔰𝔩}(2,ℂ)$, and also of their generators. Our analysis is based on the use of partial Fourier tranform applied to the realization of twisted Verma modules as $D$-modules on the Schubert cells in the full flag manifold for $\mathrm{SL}(3,ℂ)$.