Around the socle filtration of a Verma module

Question

Work in the context of a finite dimensional simple Lie algebra over $\mathbb{C}$. Write $W$ for the Weyl group and $\leq$ for the Bruhat order. For $w\in W$ let $\Delta_w$ denote the Verma module of highest weight $w^{-1}w_0\cdot 0$, where $w_0$ is the longest element in $W$ and $\cdot$ denotes the so-called dot action. So $\Delta_e$ is simple and there is a unique (up to scaling) inclusion $\Delta_v \hookrightarrow \Delta_w$ whenever $v\leq w$. In particular, there is an inclusion $\Delta_v \hookrightarrow \Delta_{w_0}$ for all $v\in W$.

For a module $M$, let $0\subset soc^1M \subset soc^2 M \subset \cdots$ denote the socle filtration. Set $soc_iM = soc^i M/soc^{i-1}M$. For example, in the case of $\mathfrak{sl}_2$: $W=\{e,s\}$ and $soc_1\Delta_s = \Delta_e = L_e$, $soc_2 \Delta_s = L_s$, where $L_w$ denotes the unique simple quotient of $\Delta_w$.

The basic question: Is it true that

a) $soc_1(\Delta_{w_0}/\Delta_w) \subseteq soc_k(\Delta_{w_0})$,

where $k$ is the smallest integer such that $soc^k(\Delta_{w_0})\not\subseteq \Delta_w$.

Why this has hope of being true: 1) It is true in type $A_1$ and $A_2$, I have not checked type $G_2$ yet out of sheer laziness (and hope that an expert will just point me to some place in the literature or tell me that I am overthinking matters). 2) It is true for $w=e$. 3) Miracles sometimes occur in Schubert varieties.

Why this has no hope of being true: 1) It has a sort of ridiculous feel to it (apologies for the cavalier attitude, but perhaps it will be justified by what follows): roughly the statement is saying that simples in Vermas can't move down in the layers of the socle filtration upon quotienting out by Verma submodules. This feels a bit nutty to me. 2) The examples of type $A_1$ and $A_2$ are both multiplicity one situations (ala occurences of simples in Vermas) and aren't really indicative of the general situation.

Some reformulations/related tidbits (that I am aware of but don't see how to leverage into a counterexample or proof):

1') The radical filtration on a Verma coincides (up to shift) with the socle filtration.

1) The statement is equivalent to the socle filtration on $\Delta_{w_0}/\Delta_w$ coinciding (up to shift) with the weight filtration (ala mixed sheaves or graded category $\mathcal{O}$), since the weight filtration on $\Delta_{w_0}$ coincides with the socle filtration (see "Proof of Jantzen conjectures" by Beilinson-Bernstein or "Koszul duality patterns in representation theory" by Beilinson-Ginzburg-Soergel). Note: the radical filtration on $\Delta_{w_0}/\Delta_w$ does coincide with the weight filtration.

2) The statement implies the assertion obtained by replacing $w_0$ with any $x$ such that $w\leq x$, since in this situation $\Delta_w\hookrightarrow \Delta_{w_0}$ factors as $\Delta_w \hookrightarrow \Delta_x \hookrightarrow \Delta_{w_0}$.

3) The question is motivated by trying to understand an analogous question for the anti-dominant projective in category $\mathcal{O}$ (principal block). Namely, let $P_e$ be the (canonical) indcomposable projective cover of $\Delta_e$ (note: $P_e$ is an amazing object, it is self-dual, injective, tilting). Are the following statements true:

b) $soc_1(P_e/\Delta_{w_0}) \subseteq soc_k(P_e)$,

where $k$ is the smallest integer such that $soc^k(P_e)\not\subseteq \Delta_{w_0}$.

c) Same question as b) but replace $w_0$ by arbitrary $w\in W$. This is of course related to a).

Added later: c) is undoubtedly false, as indicated by Dag's counterexample in type $A_1\times A_1$ below.

Added later: c) is also false for type $A_2$.

Here is why one might care: from the short exact sequence

$0\to \Delta_{w_0} \to P_e \to P_e/\Delta_{w_0} \to 0$

one deduces $Ext^1(\Delta_e, \Delta_{w_0}) = Hom(\Delta_e, P_e/\Delta_{w_0})$. Consequently, the purity (ala mixed sheaves/graded category $\mathcal{O}$) of this $Ext^1$ is (unless I am being screwy) equivalent to b).

Unless I am completely misunderstanding things, V. Mazorchuk proves this latter purity statement (in slightly different language) in Theorem 32 of http://arxiv.org/abs/math/0607589.

In fact, Theorem 32 states that $Ext^1(\Delta_v, \Delta_{w_0})$ is pure for arbitrary $v$. Now for $v=e$ this translates to b) above. This is starting to smell like a proof/answer to my questions. However, the problem is that Mazorchuk's proof (which I don't understand very well) seems to be using statements along these lines.

Related also is the fact that granted the purity of $Ext^1(\Delta_v,\Delta_{w_0})$ a downwards induction gives purity of $Ext^1(\Delta_v, \Delta_w)$. This in turn implies that the dimension of these $Ext^1$'s is given by the coefficient of $q$ (modulo sign) in the corresponding Kazhdan-Lusztig $R$-polynomial (these statements start getting me really worried, since they are certainly not true for all $Ext^i$ thanks to Boe's "Counterexample to the Gabber-Joseph conjecture").

Needless to say I am playing fast and loose with a number of things. So the assertions above should be treated with a healthy dose of suspicion (I would be grateful though to people pointing out the errors of my ways).

This of course ties in with a number of toy questions that have been bugging me:

Morphisms between Verma modules

A cohomology computation request.

Having typed all that, I really hope I didn't make a silly mistake right in the beginning!

Answer 1

Concerning the basic question (a), my first reaction is to be skeptical. Though as you say miracles sometimes do occur in this subject. I don't have a counterexample at my fingertips. Beyond the simplest cases it's extremely difficult to specify all the socle structures involved in a "typical" example.

Looking at the question in the earlier algebraic setting where Jantzen already found some miracles in the years before the Kazhdan-Lusztig Conjecture was formulated and proved, the difficult general question is what can be said about socle (or radical) filtrations of various quotients of Verma modules. It's reasonable to study first the principal block of category $\mathcal{O}$ (then consider translation functors and non-integral weights). But even here the most natural quotients are hard to analyze. There are for instance the parabolic (or "generalized") Verma modules, which show anomalies and remain only partly understood.

Your example involves the quotient of a Verma module by a proper Verma submodule, where I think most of the natural questions remain unanswered. Rather than focus at first on rank 2 simple Lie algebras, it's probably a safer option to consider instead the special case of your (a) in which $w= s_\alpha w_\circ$ for an arbitrary positive root $\alpha$ (using your notation). While developing his ingenious algebraic methods in the late 1970s, Jantzen looked closely at this special case in the final chapter of what became his Habilitationsschrift (Springer Lecture Notes in Math. 750, 1979). See especially the summary and open questions in 5.16-5.17. Though all of this predates the geometric approach using "weight" filtrations, his results show even for this simplest Verma module quotient a lot of subtlety. So it's cautionary, but might also be profitable to translate into the geometric language. Especially if you really think (a) might be true in general.

Answer 2

Here is a counter-example to (c) for the semi-simple, but not simple, algebra $\mathfrak{so}_4$.

Projectives and Vermas are described in [Brüstle, Th.; König, S.; Mazorchuk, V. The coinvariant algebra and representation types of blocks of category $\scr O$. Bull. London Math. Soc. 33 (2001), no. 6, 669--681].

The self-dual projective with their notation looks like $$\begin{matrix} && d &&\\ & b && c\\ d && a && d\\ & c && b\\ && d &&\\ \end{matrix}$$

(Here $$\begin{matrix} d &&&&\\ & c &&&\\ && d &&\\ \end{matrix}$$ is a submodule, as can be seen from the quiver presentation given in that paper.)

If we quotient out $\Delta_c$, then the radical filtration looks like $$\begin{matrix} && d &&\\ & b && c\\ d && a && d\\ &&& b &\\ \end{matrix}$$ so the lengths of radical layers in the quotient are $1$ $2$ $3$ $1$. But the socle of this quotient has length $2$ (the left-most 'd' belongs to the socle), so the socle layers are different from the radical layers in this example.