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!

`$w$`

" should be "for all`$v$`

". More important, it's risky to rely on examples of rank 2, where K-L polynomials for the (dihedral) Weyl group are trivial. Early work of people like Jantzen, Gabber-Joseph, Irving shows how delicate the socle and Ext questions can be; so it's important to focus your own questions as tightly as possible. It might also be ueful to consult Mazorchuk directly. $\endgroup$`$\mathfrak{so}_4$`

and $\Delta_w$ is one of the length $2$ Vermas. On the other hand`$\mathfrak{so}_4$`

isn't simple. Maybe this could be used to get a counterexample for`$\mathfrak{so}_6$`

? $\endgroup$2more comments