Hello,


Could anyone give a reference or proof for the following fact (which is, probably, not very difficult):


We work in category O for a semisimple complex Lie algebra. $M\_{\chi}$ denotes the Verma module with shifted highest weight $\chi$.


Fact: Let $\chi$ be dominant, $\lambda$ be such that $\chi - \lambda$ is integral. Then there exists a finite dimensional module $E$ and a surjection $E \otimes M\_{\chi} \to M\_{\lambda}$.


It seems that one should first "move" from $\chi$ to a weight in the same Weyl orbit as $\lambda$, and then...


Thanks,
Sasha



Covering of Verma modules by translation of a dominant Verma module
This is a consequence of Theorem 3.3 in Gelfand and Gelfand's [Tensor products of finite and infinite dimensional representations of semisimple Lie algebras](http://archive.numdam.org/ARCHIVE/CM/CM_1980__41_2/CM_1980__41_2_245_0/CM_1980__41_2_245_0.pdf); the projective cover of $M\_{\lambda}$ is obtained by translating $M\_\chi$ by an indecomposable projective functor.


Here's my "by hand" proof, which I'll leave here, since I wrote it out before bothering to look up the reference: The important point here is that $E\otimes M\_\chi$ has a Verma filtration where $M\_{\chi+\nu}$ appears with the multiplicity of the weight $\nu$ in $E$. The highest weight is at the bottom of the filtration, the lowest at the top. 


Consider the case when $\lambda$ is dominant; this follows by the usual argument that translation functors give equivalences between blocks of category $\mathcal{O}$; Let $E$ be a f.d. representation with $\lambda-\chi$ extremal. Thus, $M\_\lambda$ appears in the filtration on $E\otimes M\_\chi$, and no other Vermas in the same block appear, so it's a summand and thus a quotient.


Thus, it suffices to replace $\chi$ with an arbitrary dominant weight; in particular, we can assume that $\chi-\lambda$ is dominant. Now choose $E$ to be the representation with lowest weight $\lambda-\chi$. Then $M\_\lambda$ is the quotient of $E\otimes M\_\chi$ by the submodule generated by all vectors of weight $>\lambda$. Thus we're done.


