Correspondence between Verma module morphisms and invariant differential operators - is it exact?

For a (complex, connected, simply connected, simple) Lie group $G$, a parabolic subgroup $P \subseteq G$, and a $\mathfrak g$-integral $\mathfrak p$-dominant weight $\lambda$, we can construct homogeneous vector bundle $G \times_P E_{\mathfrak p}(\lambda)$ over $G/P$ (here $-\lambda$ is lowest weight) and its sheaf of holomorphic sections $\mathcal O_{\mathfrak p}(\lambda)$.

On the algebraic side, we can construct generalized Verma module $M_{\mathfrak p}(\lambda)=U(\mathfrak g) \otimes_{\mathfrak p}F_{\mathfrak p}(\lambda)$ (where $\lambda$ is highest weight).

It us well known and that there is $1-1$ correspondence between $G$-invariant differential operators $\mathcal O_{\mathfrak p}(\lambda) \to \mathcal O_{\mathfrak p}(\mu)$ and $\mathfrak g$-homomorphisms $M_{\mathfrak p}(\mu) \to M_{\mathfrak p}(\lambda)$, which is functorial. See e.g. Eastwood-Rice: Conformally invariant differential operators... or section 11.2 from the book Baston-Eastwood: The Penrose transform....

Question: Does this correspondence preserve exactness? Or maybe in a weaker form, does exact sequence of homomorphisms correspond to a sequence of differential operators exact over an affine set (big cell)? Is anything known about this, at least in $|1|$-graded (Hermitian) cases?

For example BGG resolutions of Verma modules correspond to the generalized de Rham sequences, which are locally exact (and of course, exact over an affine set).

Sketch of the correspondence: Invariant differential operator $\mathcal O_{\mathfrak p}(\lambda) \to \mathcal O_{\mathfrak p}(\mu)$ is given by a bundle morphism $J^k({\mathcal O}_{\mathfrak p}(\lambda)) \to {\mathcal O}_{\mathfrak p}(\mu)$, where $J^k$ denotes the $k$-th jet bundle. By restricting it to a fibre, and taking duals, we get a $\mathfrak p$-homomorphism $F_{\mathfrak p}(\mu) \to U^k({\mathfrak u}^-) \otimes_{\mathfrak p} F_{\mathfrak p}(\lambda) \subseteq M_{\mathfrak p}(\lambda)$, where $U^k({\mathfrak u}^-)$ denotes the $k$-th filtered piece of the universal enveloping algebra of the opposite nilpotent radical of $\mathfrak p$. By Frobenius reciprocity (adjoint functors), we get $\mathfrak g$-homomorphism $M_{\mathfrak p}(\mu) \to M_{\mathfrak p}(\lambda)$. This process is reversible.

• Can you, please, carefully define what the terms "functorial" and "preserve exactness" mean in the context of your question? As stated, they do not quite make sense (for example, a sequence of Verma modules is never exact). – Victor Protsak Mar 15 '17 at 6:28
• Functorial: Composition $M_{\mathfrak p}(\nu) \stackrel{f}{\to} M_{\mathfrak p}(\mu) \stackrel{g}{\to} M_{\mathfrak p}(\lambda)$ corresponds to the composition $\mathcal O_{\mathfrak p}(\lambda) \stackrel{D_g}{\to} \mathcal O_{\mathfrak p}(\mu) \stackrel{D_g}{\to} \mathcal O_{\mathfrak p}(\nu)$, where $D_g$ and $D_f$ correspond to $g$ and $f$, and of course $D_1=1$. – Rafael Mrđen Mar 15 '17 at 9:01
• I realize that "preserve exactness" is not completely clear. For start, can we claim that $M_{\mathfrak p}(\mu) \stackrel{g}{\to} M_{\mathfrak p}(\lambda)$ is monomorphism iff corresponding $\mathcal O_{\mathfrak p}(\lambda) \stackrel{D_g}{\to} \mathcal O_{\mathfrak p}(\mu)$ is locally surjective, or perhaps surjective on sections over big affine cell? – Rafael Mrđen Mar 15 '17 at 9:07
• More generally, for $M_{\mathfrak p}(\nu) \stackrel{f}{\to} M_{\mathfrak p}(\mu) \stackrel{g}{\to} M_{\mathfrak p}(\lambda)$ such that $Im(f)=Ker(g)$ (surely such exist), does the same hold for operators (locally or on affine set)? – Rafael Mrđen Mar 15 '17 at 9:13
• And finally, what about BGG-like sequences? I mean, if I have direct sums of Vermas and and their homomorphisms $\bigoplus_\nu M_{\mathfrak p}(\nu) {\to}\bigoplus_\mu M_{\mathfrak p}(\mu) {\to} \bigoplus_\lambda M_{\mathfrak p}(\lambda)$, exact in the middle, would the corresponding direct sum of differential operators have the same property? – Rafael Mrđen Mar 15 '17 at 9:22