First, let me fix some notation. Let $\mathfrak{g}\_1$ be a semisimple Lie algebra and let $\mathfrak{p}$ be its parabolic subalgebra which induces the grading $\mathfrak{g} = {\mathfrak{g}}\_{-} \oplus {\mathfrak{g}}\_{0} \oplus {\mathfrak{g}}\_{+}$, where $\mathfrak{g}\_{0}$ is the Levi part of $\mathfrak{p}$ and $\mathfrak{g}\_+$ its nilpotent radical. Let $\lambda$ be a weight of an irreducible finite dimensional $\mathfrak{p}$-representation $F_\lambda$ and let $V_\lambda$ be the generalized Verma module $\mathfrak{U_g}\otimes_{\mathfrak{U_p}} F_\lambda$.

There is an article^{1}, which suggest that the invariant operators between associated bundles over $G/P$ are given by obstruction in cohomology $H^k(\mathfrak{g}\_-,V_\lambda)$, at least in the 1-graded case.

Is this result known and if yes is there a more mathematical treatment?

What is known about the cohomological groups $H^k(\mathfrak{g}\_-,V_\lambda)$? (Results of Collingwood & Boe for the 1-graded case are referenced in the mentioned article, but they are 25 years old.)

- The article is "Unfolded form of conformal equations in M dimensions and o(M+2)-modules" by O.V. Shaynkman, I.Yu. Tipunin and M.A. Vasiliev and it was published in Rev. Math. Phys.