Bruhat ordering and non-vanishing Extension groups

Question

Let $P_{x,w}(q)$ be the Kazhdan Lusztig polynomial. It is well-known that $P_{x,w}(q)\neq 0\iff x\le w$.

By the interpretation of the Kazhdan Lusztig polynomial in terms of extension group, it holds that $x\le w \iff \mathrm{Ext}_{\mathcal{O}}^i(M(x\cdot(-2\rho)),L(w\cdot(-2\rho)))\neq 0$ for some $i\ge 0$, where $M(\eta)$ is the Verma module, $L(\eta)$ is its unique simple quotient and $\rho$ is the half sum of positive roots.

Let $\mu$ be an integral, antidominant weight, $\Delta$ be the set of simple roots, $\Sigma=\{\alpha\in\Delta:\langle\mu+\rho,\alpha^\lor\rangle=0\}$, $W^{\Sigma}=\{w\in W:w<ws_\alpha,\forall \alpha\in\Sigma_\mu\}$ and $x,w\in W^{\Sigma}$.

It is not hard to show $\mathrm{Ext}_{\mathcal{O}}^i(M(x\cdot\mu),L(w\cdot\mu))\neq 0$ for some $i\ge 0\implies x\le w$.

I would like to know whether the converse:

$x\le w\implies \mathrm{Ext}_{\mathcal{O}}^i(M(x\cdot\mu),L(w\cdot\mu))\neq 0$ for some $i\ge 0$

is true or not. If it is true, I would like to know why. If not, any counter-example?

Answer 1

It is not true in general. Take $\mathfrak{sl}_3$, with simple reflections $s,t$, such that $s$ is singular. Put $x=e$, $w= st$; both are in $W^\Sigma$.

I claim that $Ext_\mathcal{O}^i(M(\mu),L(st \cdot \mu)) = 0$ for all $i \geq 0$.

Denote by $\lambda$ a regular, integral antidominant weight. By [2, Corollary 1.3.3.] we have

$$ \dim Ext_\mathcal{O}^i(M(\mu),L(st \cdot \mu)) = \begin{cases} \dim Hom(M(\lambda),L(st \cdot \lambda))=0 & \colon i=0 \\ \dim Ext_\mathcal{O}^i(M(\lambda),L(st \cdot \lambda)) - \dim Ext_\mathcal{O}^{i-1}(M(s \cdot \lambda),L(st \cdot \lambda)) &:i>0 \end{cases}$$.

We know that $P_{e,st} = P_{s,st} = 1$. From KL polynomial in terms of extension groups in the regular case [1, Theorem 8.11.], it follows that the above always cancels. (For $i=2$ we have $1-1=0$.)

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[1] Humphreys: Representations of semisimple Lie algebras in the BGG category O. Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI, 2008.

[2] Irving: Singular blocks of the category O, Math Z (1990) 204: 209.

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EDIT: General explanation: A singular block of $\mathcal{O}$ is Koszul-dual to the regular block of the parabolic category $\mathcal{O}^{\mathfrak{p}}$, where $\mathfrak{p}$ is given by the same singularity set. By this duality, the dimensions of Ext-group correspond to the multiplicities of simples inside generalized Verma modules. But strange things may happen with simples inside generalized Vermas, in particular, some simples that are "expected" to appear by the Bruhat order, may not appear. This is because a generalized Verma is a quotient of the usual Verma, where you kill all the submodules that are not in $\mathcal{O}^{\mathfrak{p}}$. But a lot of composition factors that actually exist in $\mathcal{O}^{\mathfrak{p}}$ are killed in the quotient.