Covering of Verma modules by translation of a dominant Verma module

Question

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

Answer 1

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; 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.

Answer 2

Probably there is no explicit reference for this statement (assuming it's true), but I should emphasize that it doesn't seem to involve Jantzen's translation functors. Instead you are starting with a more general problem of understanding how a given (dominant) Verma module tensors with a finite dimensional module. This is usually very complicated to analyze, since it involves filtrations in which every quotient is a Verma module. Splitting into direct summands is not easy to predict, apart from use of the linkage principle. Some of the Verma modules in a filtration do occur as quotients, some not.

Here you assume the Verma module has dominant highest weight, which adds more structure: such Verma modules are among the projective objects in the category $\mathcal{O}$. I'm not sure whether you assume that $\chi$ is itself an integral weight, but that's the easiest case to start with. Then $\lambda$ is also integral. So the difference $\chi - \lambda$ is conjugate under the Weyl group $W$ to a unique dominant integral weight, which is the highest weight of a finite dimensional module possibly suitable for your purposes.

Your notation follows the Moscow/Paris convention of the 1970s, but I prefer Jantzen's more straightforward set-up. This uses the dot action of $W$, with $w \cdot \chi = w(\chi + \rho) - \rho$. He denotes a Verma module of highest weight $\chi$ by $M(\chi)$ and its unique simple quotient by $L(\chi)$, etc. My graduate text Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$ (AMS, 2008) uses more modern arguments for BGG reciprocity and the like, but follows at first Jantzen's 1979 Springer Lecture Notes. The basic ideas about tensor products $M(\chi) \otimes L(\mu)$ (with $\mu$ dominant integral) are discussed in my 3.6; note especially the last two paragraphs. More restrictive hypotheses are imposed in Chapter 7, where the effect of a translation functor on a Verma module is computed in 7.6.

Anyway, your assumptions lead to a direct sum of various indecomposable projectives when you tensor with a finite dimensional module, so this is the starting point for what you need to study. But I'd have to look more carefully at low rank examples to estimate what choice of $\mu$ could lead to the quotient $M(\lambda)$ you want: its highest weight $\lambda$ might or might not be dominant, and the filtration of the tensor product quickly gets complicated.

ADDED: As Ben points out, the paper by Bernstein-Gelfand on projective functors may provide a substitute for the direct use of translation functors here. But it does get complicated, since the method of Bernstein-Gelfand goes outside the framework of category $\mathcal{O}$. Their paper is not easy reading in any case. Some of the significant aspects of their classification theorem are formulated in sections 10.5-10.8 of my book (I hope correctly).