BGG complexes in singular blocks of category O

TL;DR

This paper extends the theory of BGG complexes to singular blocks of category O, providing criteria for their exactness and constructing resolutions using Kazhdan-Lusztig-Vogan polynomials and generalized Verma modules.

Contribution

It introduces new criteria for the exactness of BGG complexes in singular blocks and generalizes the construction to balanced quasi-hereditary algebras.

Findings

Criteria for BGG complex exactness using Kazhdan-Lusztig-Vogan polynomials

Construction of BGG complexes in balanced quasi-hereditary algebras

Resolutions of simple modules in subcategories of category O

Abstract

Using translation from the regular block, we construct and analyze properties of BGG complexes in singular blocks of BGG category ${\mathcal{O}}$. We provide criteria, in terms of the Kazhdan-Lusztig-Vogan polynomials, for such complexes to be exact. In the Koszul dual picture, exactness of BGG complexes is expressed as a certain condition on a generalized Verma flag of an indecomposable projective object in the corresponding block of parabolic category ${\mathcal{O}}$. In the second part of the paper, we construct BGG complexes in a more general setting of balanced quasi-hereditary algebras and show how our results for singular blocks can be used to construct BGG resolutions of simple modules in ${\mathcal{S}}$-subcategories in ${\mathcal{O}}$.

Equations

$$u=x_0\to x_1\to \dots \to x_k=v.$$

$$l(x)<l(v^{\lambda })\le l(v^{\lambda })+l(v_{\lambda })=l(sx)=l(x)+1,$$

$$y^{\prime \prime }<s{y}^{\prime \prime }\text{ and }z<sz,$$

$$s{y}^{\prime \prime }<y^{\prime \prime }\text{ and }z<sz,$$

$$\mu (w,x)=\{\begin{array}{ll}1,& \text{if }w=x;\\ -\sum _{w\le z<x}\mu (w,z),& \text{if }w<x.\end{array}$$

$${\mu }^{\lambda }(w,x)=\{\begin{array}{ll}0,& \text{if there exists }z\notin {\tilde{W}}^{\lambda }\text{ such that }w<z<x;\\ (-1{)}^{l(x)-l(w)},& \text{otherwise}.\end{array}$$

$$X_w^i:=\left\{x\in {\tilde{W}}^{\lambda }:w\le x,{\mu }^{\lambda }(w,x)\ne 0\text{ and }l(x)=l(w)+i\right\}.$$

$$\dots \to \underset{x\in X_w^{i+1}}{⨁}\Delta (x\cdot \lambda )\to \underset{x\in X_w^i}{⨁}\Delta (x\cdot \lambda )\to \dots \to \Delta (w\cdot \lambda )\to L(w\cdot \lambda )\to 0.$$

$$\dots \to \underset{\begin{array}{c}x\ge w\\ l(x)=i+1\end{array}}{⨁}\Delta (x\cdot 0)\to \underset{\begin{array}{c}x\ge w\\ l(x)=i\end{array}}{⨁}\Delta (x\cdot 0)\to \dots \to \Delta (w\cdot 0)\to L(w\cdot 0)\to 0.$$

$$T_0^{\lambda }:{\mathcal{O}}_0\to {\mathcal{O}}_{\lambda }$$

$$A\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.75pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&\crcr}}}\ignorespaces{\hbox{\kern-6.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 30.75pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 68.83679pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 68.83679pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 84.80443pt\raise 6.1111pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.75pt\hbox{$\scriptstyle{f-hg}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 106.69928pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 106.69928pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 145.25623pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 145.25623pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 183.21454pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 183.21454pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{F}$}}}}}}}\ignorespaces}}}}\ignorespaces.$$

$$\Delta (z\cdot \lambda )\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Delta (y_2\cdot \lambda )\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\Delta (x_4\cdot \lambda )\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$w:=w_0^{\lambda }=s_3{s}_2{s}_3{s}_2\in {\tilde{W}}^{\lambda }.$$

$$v:=s_2{s}_3{s}_2{s}_1{s}_2{s}_3{s}_2.$$

$$P_{w,x}(q)=\{\begin{array}{ll}1+q,& \text{ if }x=v;\\ 1,& \text{ if }x\ge w,x\ne v.\end{array}$$

$$\dots \to \underset{x\in A}{⨁}\Delta (x\cdot 0)\stackrel{d_1}{⟶}\Delta (w\cdot 0)\stackrel{d_0}{\twoheadrightarrow }L(w\cdot 0)\to 0.$$

$${\tilde{W}}^{\lambda }=\{w\to s_{1}w\to s_2{s}_{1}w\to s_3{s}_2{s}_{1}w\to s_2{s}_3{s}_2{s}_{1}w\to s_1{s}_2{s}_3{s}_2{s}_{1}w\}.$$

$$0\to \Delta (s_{1}w\cdot \lambda )\hookrightarrow \Delta (w\cdot \lambda )\twoheadrightarrow L(w\cdot \lambda )\to 0.$$

$$P_{y,z}^{\mu }(q)=\sum _j\dim {\mathrm{Ext}}_{\mathcal{O}}^j(\Delta (y\cdot \mu ),L(z\cdot \mu ))\cdot q^{\frac{l(z)-l(y)-j}{2}}.$$

$$P_{y,z}^{\mu }=\sum _{t\in W_{\mu }}(-1{)}^{l(t)}{P}_{yt,z}.$$

$$P_{x{w}_0,w{w}_0}^{w_0\cdot \lambda }(q)=\sum _j\dim {\mathrm{Ext}}_{\mathcal{O}}^j(\Delta (x\cdot \lambda ),L(w\cdot \lambda ))\cdot q^{\frac{l(x)-l(w)-j}{2}}.$$

$$H^i\left({\mathfrak{n}}^{+},L(w\cdot \lambda )\right)=\underset{x\in X_w^i}{⨁}{\mathbb{C}}_{x\cdot \lambda },$$

$$\dim {\mathrm{Ext}}_{\mathcal{O}}^i\left(\Delta (x\cdot \lambda ),L(w\cdot \lambda )\right)=\{\begin{array}{ll}1& :x\in X_w^i\\ 0& :\text{otherwise}.\end{array}$$

$$H^i\left({\mathfrak{n}}^{+},L(w\cdot \lambda )\right)\cong H_i\left({\mathfrak{n}}^{-},L(w\cdot \lambda )\right),$$

$$\hat{w}:=w^{-1}{w}_0.$$

$$\left(P(\xi ):\Delta (\nu )⟨i⟩\right)=\left[\Delta (\nu ):L(\xi )⟨i⟩\right]=\left[{\mathrm{rad}}^i\Delta (\nu ):L(\xi )\right].$$

$$\left(P(w\cdot \lambda ):\Delta (x\cdot \lambda )⟨i⟩\right)=\{\begin{array}{ll}1,& \text{if }i=l(w)-l(x);\\ 0,& \text{otherwise.}\end{array}$$

$$\begin{array}[]{ccc}&{\Delta}^{\lambda}(s_{2}s_{3}s_{1}s_{2}\cdot 0)&\\ {\Delta}^{\lambda}(s_{2}s_{3}s_{1}\cdot 0)\langle 1\rangle&&\ {\Delta}^{\lambda}(s_{2}\cdot 0)\langle 1\rangle\\ &{\Delta}^{\lambda}(e\cdot 0)\langle 2\rangle.&\end{array}$$

Full text

In the second part of the paper, we construct BGG complexes in a more general setting of balanced quasi-hereditary algebras and show how our results for singular blocks can be used to construct BGG resolutions of simple modules in ${\mathcal{S}}$-subcategories in ${\mathcal{O}}$.

1. Introduction and description of the results

The classical BGG resolution from [BGG75] is a resolution of a simple finite dimensional module over a semi-simple complex finite dimensional Lie algebra in terms of Verma modules. It has many applications, for example, it can be used to compute all self-extensions of this simple finite dimensional module inside BGG category $\mathcal{O}$, see [BGG76]. Various generalizations and analogues of BGG resolution were studied in many different contexts, see, e.g., [Le77, RC80, BH09, EH04, FM98, Kö95, Co14]. Some geometric constructions of BGG complexes in the setting of homogeneous bundles and invariant differential operators (which is, in a certain sense, dual to category $\mathcal{O}$) were described in [ČSS01, PS17, Mr17, HM18].

One natural generalization is to consider resolutions of finite dimensional simple modules using “standard” modules in other categories, for example in the parabolic version of $\mathcal{O}$. This question was studied in [Le77]. Another natural generalization is to ask which other simple module in $\mathcal{O}$ have resolutions by Verma modules. For modules inside the regular block of $\mathcal{O}$, this question was studied in [BH09].

The aim of the present paper is to investigate which simple modules in singular blocks of $\mathcal{O}$ have resolutions by Verma modules. We give a combinatorial answer involving Möbius function for the poset of shortest coset representatives in the cosets of a Weyl group modulo the parabolic subgroup associated with the singularity and Kazhdan-Lusztig-Vogan polynomials. The answer is explicit enough to be verifiable by a computer, so we provide the lists of such modules in small ranks (or, more precisely, we provide lists of modules which do not have this property as in small ranks the number of the latter modules is significantly smaller).

Our results also have applications to construction of BGG type resolutions for non-quasi-hereditary generalizations of category $\mathcal{O}$ studied in [FKM00]. In fact, our results provide a complete answer for existence of BGG resolutions of simple modules in the regular block for such categories. Using the equivalence in [MS08], this implies existence of BGG-type resolutions for very general setups of parabolically induced modules.

The paper is organized as follows. We describe our basic setup in Section 2. In Section 3, we collect some auxiliary statement about combinatorics of Bruhat order on Weyl groups. In Section 4 we define singular BGG complexes and we study their exactness, in terms of Kazhdan-Lusztig-Vogan polynomials, in Section 5. In Section 6 we connect the problem of existence of BGG resolutions with the Koszul dual picture of parabolic category $\mathcal{O}$, where the problem is reformulated in terms of Verma flags of indecomposable projective modules. This allows us to give a sufficient and necessary conditions for existence of BGG resolution in Proposition 30. In Section 7 we list results of computations in low rank case. In Section 8 we interpret BGG complexes and resolution in terms of complexes of structural modules over balanced quasi-hereditary algebras. Finally, in Section 9 we describe how our results can be applied to construct BGG resolutions for $\mathcal{S}$-subcategories in $\mathcal{O}$.

Acknowledgements: This research was partially supported by the Swedish Research Council, Göran Gustafsson Stiftelse and Vergstiftelsen. R. M. was also partially supported by the QuantiXLie Center of Excellence grant no. KK.01.1.1.01.0004 funded by the European Regional Development Fund. We thank Axel Hultman for help with Lemma 3.

2. Setup

In this paper we work over the base filed $\mathbb{C}$ of complex numbers.

We let $\mathfrak{g}$ denote a semi-simple finite dimensional Lie algebra with a fixed triangular decomposition ${\mathfrak{g}}={\mathfrak{n}}^{-}\oplus{\mathfrak{h}}\oplus{\mathfrak{n}}^{+}$. Associated to such a triangular decomposition, we have the corresponding BGG category $\mathcal{O}$, see [BGG76, Hu08].

For $\lambda\in\mathfrak{h}^{*}$, we denote by ${\Delta}(\lambda)$ the Verma module with highest weight $\lambda$. The simple quotient of ${\Delta}(\lambda)$ is denoted by $L(\lambda)$, and the indecomposable projective cover of $L(\lambda)$ in $\mathcal{O}$ is denoted by $P(\lambda)$.

Let $W$ denote the Weyl group of $\mathfrak{g}$ which acts on $\mathfrak{h}^{*}$ in the usual way. The above triangular decomposition leads to a decomposition of the root system of $\mathfrak{g}$ into positive and negative roots and we denote by $\rho$ the half of the sum of all positive roots. Then the dot action of $W$ on $\mathfrak{h}^{*}$ is given by $w\cdot\lambda=w(\lambda+\rho)-\rho$.

Category $\mathcal{O}$ decomposes into blocks with respect to the action of the center of the universal enveloping algebra of $\mathfrak{g}$. For $\lambda\in\mathfrak{h}^{*}$, we denote by ${\mathcal{O}}_{\lambda}$ the block which corresponds to the central character of the Verma module $\Delta(\lambda)$. Thanks to Soergel’s combinatorial description of blocks of $\mathcal{O}$ from [So90], without loss of generality we may work with integral weights.

Let $\{\alpha_{1},\ldots,\alpha_{n}\}$ be a fixed (standard) ordering of simple roots. The simple reflection in $W$ corresponding to $\alpha_{i}$ is denoted $s_{\alpha_{i}}=s_{i}$.

A weight $\lambda$ is dominant if $\langle\lambda+\rho,\alpha\rangle\geq 0$ for all simple roots $\alpha$. It is regular if $\langle\lambda+\rho,\alpha\rangle\neq 0$ for all simple roots $\alpha$, otherwise it is singular.

For a subset $S\subset\{\alpha_{1},\ldots,\alpha_{n}\}$, we have the corresponding parabolic subcategory $\mathcal{O}^{S}$ in $\mathcal{O}$ as defined in [RC80]. If $\lambda\in\mathfrak{h}^{*}$ is dominant and such that its stabilizer in $W$ with respect to the dot-action is the parabolic subgroup generated by simple reflections from $S$, we will also use the notation ${\mathcal{O}}^{\lambda}:=\mathcal{O}^{S}$. The generalized Verma module in ${\mathcal{O}}^{\lambda}$ with highest weight $\mu$ is denoted ${\Delta}^{\lambda}(\mu)$ and its indecomposable projective cover in ${\mathcal{O}}^{\lambda}$ is denoted $P^{\lambda}(\mu)$.

We refer to [Hu08] for details.

3. Combinatorics of the Weyl group

3.1. Conventions and preliminaries

Consider the usual length function $l$ and the Bruhat order $<$ on $W$. Put $u\to v$ if $u<v$ and $l(v)=l(u)+1$. With respect to Bruhat order, $W$ is a graded poset with degree given by the length function, in particular, $u\leq v$ if and only if there is a path

[TABLE]

Recall that $u\leq v$ also means that some (equivalently, any) reduced expression for $v$ contains a reduced expression for $u$ as a subword. Also recall that, for a simple reflection $s$, $sv<v$ if and only if $v$ has a reduced expression that starts with $s$, moreover, in the latter case $sv$ does not have such an expression. This will be often used without mentioning. For details, see [BB06, Chapter 2].

Fix a dominant, but possibly singular weight $\lambda$, and set $S$ to be the set of simple roots $\alpha$ for which $\langle\lambda+\rho,\alpha\rangle=0$. Elements of $S$ are called singular roots, and reflections $s_{\alpha}$, for $\alpha\in S$, are called singular reflections. Singular reflections generate a (parabolic) subgroup $W_{\lambda}\leq W$, which is precisely the stabilizer of $\lambda$ with respect to the dot action. Denote by $W^{\lambda}$ the set of minimal length representatives of the cosets $W/W_{\lambda}$. It is a graded subposet of $W$. For $u,v\in W^{\lambda}$, we have $u\leq v$ if and only if there is a path (1) completely contained in $W^{\lambda}$. Recall the following standard fact.

Lemma 1** (Kostant, [Kos61, Proposition 5.13.]).**

Any $v\in W$ can be uniquely decomposed as $v=v^{\lambda}v_{\lambda}$, where $v^{\lambda}\in W^{\lambda}$ and $v_{\lambda}\in W_{\lambda}$. Moreover, $l(v)=l(v^{\lambda})+l(v_{\lambda})$.

Denote by $w_{0}$ the longest element in $W$, and decompose it via Kostant’s lemma: $w_{0}=(w_{0})^{\lambda}(w_{0})_{\lambda}$. Then $(w_{0})_{\lambda}$ is the longest element in $W_{\lambda}$, and we denote it, as usual, by $w_{0}^{\lambda}$. The set $W^{\lambda}w_{0}^{\lambda}$ is precisely the set of the longest representatives of the cosets $W/W_{\lambda}$. Since it will be frequently used, we reserve a special notation $\tilde{W}^{\lambda}:=W^{\lambda}w_{0}^{\lambda}$ for it.

3.2. Intersection of intervals and cosets

Here we prove the main auxiliary combinatorial statements that will be used in the next section.

Lemma 2**.**

Assume $x\in W^{\lambda}$ and $s$ is a simple reflection.

1. $($a$)$

If $sx\not\in xW_{\lambda}$, then $sx\in W^{\lambda}$ and the multiplication by $s$ from the left gives a directed graph isomorphism $xW_{\lambda}\leftrightarrow sxW_{\lambda}$. 2. $($b$)$

If $sx\in xW_{\lambda}$, then $sz\in xW_{\lambda}$, for all $z\in xW_{\lambda}$.

Proof.

Let us start with claim (a). Suppose first that $sx<x$. To see that $sx\in W^{\lambda}$, decompose $sx$ as $v^{\lambda}v_{\lambda}$ according to Kostant’s lemma. Then $x=sv^{\lambda}v_{\lambda}$, without any cancellation. If $v_{\lambda}$ were non-trivial, this would give a reduced expression for $x$ ending in a singular reflection, which contradicts $x\in W^{\lambda}$. So $v_{\lambda}=e$, and $sx=v^{\lambda}\in W^{\lambda}$.

If $x<sx$ and $sx\not\in xW_{\lambda}$, decompose again $sx=v^{\lambda}v_{\lambda}$ according to Kostant’s lemma. Then $x<v^{\lambda}v_{\lambda}$, but since $x$ cannot have reduced expression ending with a singular root, we must have $x<v^{\lambda}$. We have

[TABLE]

so $v_{\lambda}=e$, and $sx=v^{\lambda}\in W^{\lambda}$. Claim (a) now follows directly from Kostant’s lemma.

We proceed with claim (b). If $sx\in xW_{\lambda}$, then, by Kostant’s lemma, $sx=xt$, for some simple singular reflection $t$. Any $z\in xW_{\lambda}$ has the form $z=xu$ for some $u\in W_{\lambda}$, so $sz=sxu=xtu\in xW_{\lambda}$. This completes the proof. ∎

The next statement was suggested to us by Axel Hultman who also provided some hints about the proof.

Lemma 3**.**

Assume that $x\in W^{\lambda}$ and $w\in W$ are such that $x\leq w$. Then the intersection $[e,w]\cap xW_{\lambda}$ has a unique maximal element.

Proof.

We prove this by induction on $l(w)$. The basis of the induction $w=x=e$ is trivial. Let $w$ and $x$ be as in the statement, and choose a simple reflection $s$ such that $sw<w$. Recall that in this case $w$ has a reduced expression starting with $s$, that is $w=ss_{2}\ldots s_{l(w)}$, and thus we also have $sw=s_{2}\ldots s_{l(w)}$.

By induction, we can do the following:

• •

If $sx\leq sw$, we set $y^{\prime}$ to be the unique maximal element in $[e,sw]\cap sxW_{\lambda}$.

• •

If $x\leq sw$, we set $y^{\prime\prime}$ to be the unique maximal element in $[e,sw]\cap xW_{\lambda}$.

Using $s$, $y^{\prime}$ and $y^{\prime\prime}$, we will construct the unique maximal element $y$ in $[e,w]\cap xW_{\lambda}$. We have to distinguish between several cases.

Case 1. Assume that $sx<x$. Obviously, $sx\not\in xW_{\lambda}$, so, by Lemma 2, we have $sx\in W^{\lambda}$ and thus there is a directed graph isomorphism $sxW_{\lambda}\to xW_{\lambda}$.

From $sx<x\leq w$, it follows that $sx\leq sw$ and thus $y^{\prime}$ exists. We have $sy^{\prime}\in xW_{\lambda}$ and, since $y^{\prime}\leq sw$, we have $sy^{\prime}\leq w$. So $y:=sy^{\prime}$ is in the intersection $[e,w]\cap xW_{\lambda}$. We want to see that $y$ is the unique maximum element in this intersection.

Take any $z$ in $[e,w]\cap xW_{\lambda}$. Then $sz\in sxW_{\lambda}$ and, since $sz<z\leq w$, we have $sz\leq sw$. By the inductive assumption, $sz\leq y^{\prime}$ and therefore $z\leq sy^{\prime}=y$. This completes Case 1.

Case 2. Assume that $x<sx$. No reduced expression of $x$ can start with $s$, so, in this case, we have $x\leq sw$. Therefore, $y^{\prime\prime}$ exists by the inductive assumption. Note that $y^{\prime\prime}\leq w$.

Subcase 2a. Assume that $sx\not\in xW_{\lambda}$. By Lemma 2, we have a directed graph isomorphism $xW_{\lambda}\to sxW_{\lambda}$. In particular, no element in $xW_{\lambda}$ can have a reduced expression that starts with $s$. Consequently, if $z\in xW_{\lambda}$ with $z\leq w$, then $z\leq sw$. Therefore $z\leq y^{\prime\prime}$, and thus $y:=y^{\prime\prime}$ is the element we are looking for.

Subcase 2b. Assume that $sx\in xW_{\lambda}$. By Lemma 2, we have $sy^{\prime\prime}\in xW_{\lambda}$. Note that $sy^{\prime\prime}\leq w$. In this case we define $y:=\max\{y^{\prime\prime},sy^{\prime\prime}\}$. Take $z\in xW_{\lambda}$ with $z\leq w$. Then also $sz\in xW_{\lambda}$. We either have $z\leq sw$ or $sz\leq sw$. By induction, we either have $z\leq y^{\prime\prime}\leq y$ or $sz\leq y^{\prime\prime}$. In the former case we are done. In the latter case there are four possibilities:

[TABLE]

For each of them, it is straightforward to check that $z\leq y$. ∎

Remark 4**.**

The special case $x=e$ of the previous lemma is proved in [vdH74, Lemma 7].

Lemma 5**.**

Assume that $x\in W^{\lambda}$ and $w\in W$ are such that $w\leq x$. Then the intersection $[w,w_{0}]\cap xW_{\lambda}$ has a unique minimal element. Moreover, the intersection $[w,w_{0}]\cap xW_{\lambda}$ is isomorphic, as a directed graph, to the interval $[y,w_{0}^{\lambda}]$ in $W_{\lambda}$.

Proof.

The first claim follows from Lemma 3 by multiplication with $w_{0}$. The second claim follows from Kostant’s lemma. Indeed, $y$ can be taken as the $W_{\lambda}$-component of the unique minimum from the intersection. ∎

Lemma 6**.**

Assume that $x\in W^{\lambda}$ and $w\in W$ are such that $w\leq x$. Assume further that the intersection $[w,w_{0}]\cap xW_{\lambda}$ is not a singleton. Then there is a partition ${\mathcal{P}}$ of $[w,w_{0}]\cap xW_{\lambda}$ consisting of $2$-subsets such that:

1. $($a$)$

For any $\{z,z^{\prime}\}\in{\mathcal{P}}$, there is an arrow $z\to z^{\prime}$ or $z^{\prime}\to z$. 2. $($b$)$

If both $\{z\to z^{\prime}\}$ and $\{t\to t^{\prime}\}$ are in ${\mathcal{P}}$ and $l(z)=l(t)$ but $z\neq t$, then $z\not\leq t^{\prime}$.

Proof.

Because of the second claim in Lemma 5, it is enough to prove the statement for any interval $[y,w_{0}]\subseteq W$, where $y<w_{0}$.

Take any simple reflection $s$ such that $y<sy$. The existence of such $s$ is guaranteed by [BB06, Proposition 2.3.1]. We claim that the wanted partition can be given by $\{z,sz\}$, where $z\in[y,w_{0}]$.

To see that this partition has property (a) we need to check the following: if $y<z$ and $sz<z$, then $y\leq sz$. But this follows directly from the subword property.

To see that this partition has property (b), suppose $z<sz$, $t<st$, and $z\leq st$. Then, by the subword property, we have $z\leq t$. But, if, in addition, we suppose that $l(z)=l(t)$, we obtain $z=t$. The claim follows. ∎

3.3. Möbius function

Recall that each locally finite partially ordered set has its Möbius function $\mu$, defined as the inverse, in the incidence algebra, of the defining $\zeta$-function of the poset. For more information, see [Sta11, Chapter 3]. Concretely, the function $\mu$ can be defined, for pairs $w\leq x$, recursively:

[TABLE]

We emphasize that the value $\mu(w,x)$ depends only on the interval $[w,x]$.

The main result of [Ve71] describes the Möbius function for the Bruhat order on Weyl groups. We will need the following generalization from [BB06, Section 2.7], which describes the Möbius function for the restriction of the Bruhat order on $W^{\lambda}$, or, equivalently, on $\tilde{W}^{\lambda}$.

Proposition 7**.**

For $w,x\in\tilde{W}^{\lambda}$ with $w\leq x$, we have

[TABLE]

In other words, $\mu^{\lambda}(w,x)=0$ if and only if there exists a directed path $w\to\ldots\to x$ that exits $\tilde{W}^{\lambda}$.

Lemma 8**.**

For $w,x\in\tilde{W}^{\lambda}$ with $w\leq x$, the following assertions are equivalent:

1. $($a$)$

The value $\mu^{\lambda}(w,x)\neq 0$. 2. $($b$)$

The intersection $[w,w_{0}]\cap xW_{\lambda}$ is a singleton.

Proof.

Negation of (b) implies $\mu^{\lambda}(w,x)=0$ by Proposition 7.

Now, suppose $\mu^{\lambda}(w,x)=0$, and take $z\notin\tilde{W}^{\lambda}$ such that $w<z<x$. Decompose $z=z^{\lambda}z_{\lambda}$ and $x=x^{\lambda}x_{\lambda}$, according to Kostant’s lemma. Then $z_{\lambda}\neq w_{0}^{\lambda}$ and $x_{\lambda}=w_{0}^{\lambda}$. By [BB06, Proposition 2.5.1.], we have $z^{\lambda}\leq x^{\lambda}$. But then, again by Kostant’s lemma and the subword property, we have $z\leq x^{\lambda}z_{\lambda}<x$. This means that we have at least two different elements, $x^{\lambda}z_{\lambda}$ and $x$ in $[w,w_{0}]\cap xW_{\lambda}$. The claim follows. ∎

4. Singular BGG complexes

4.1. BGG complex

From now on we let $\lambda$ to be a dominant integral weight. Verma modules and simple modules in the (singular) block ${\mathcal{O}}_{\lambda}$ are parameterized by the set $W^{\lambda}$, or, equivalently, by the set $\tilde{W}^{\lambda}$. We will use the latter parameterization, since it agrees better with translations to walls.

In this section, to each simple module $L(w\cdot\lambda)$ in ${\mathcal{O}}_{\lambda}$, where $w\in\tilde{W}^{\lambda}$, we will attach canonically its (singular) BGG complex. So, fix $w\in\tilde{W}^{\lambda}$. Set

[TABLE]

Clearly, there exists $t\leq 0$ such that $X^{i}$ is non-empty, for $i=0,1,\ldots,t$, and empty otherwise. Set $X_{w}=X^{0}_{w}\cup\ldots\cup X^{t}_{w}$.

Consider the sequence of the form

[TABLE]

Later on we will define differentials in this sequence such that it becomes a complex, which we will call the (singular) BGG complex attached to $L(w\cdot\lambda)$. The differentials in (2) will consist of direct sums of monomorphisms ${\Delta}(x^{\prime}\cdot\lambda)\hookrightarrow{\Delta}(x\cdot\lambda)$, for $x^{\prime}\in X^{i+1}_{w}$ and $x\in X^{i}_{w}$ such that there is an arrow $x\to x^{\prime}$. These monomorphisms are determined uniquely up to a scalar. We will show that it is possible to choose these monomorphisms (defined up to scalar) such that (2) becomes a complex. We will also determine under which conditions this complex is exact, that is, is a resolution of $L(w\cdot\lambda)$.

Remark 9**.**

The necessity of using $X_{w}$ as the index set of a BGG complex follows from the Möbius inversion formula applied to the Euler characteristic of the BGG complex in the graded Grothendieck group, see Proposition 29 for details.

4.2. Translation to the wall

The first step in our construction is to consider the classical BGG resolution of $L(w\cdot 0)$ in ${\mathcal{O}}_{0}$, see [BH09, Subsection 4.2.],

[TABLE]

Let us translate the complex in (3), which we denote by $\boldsymbol{\Delta}$, to the $\lambda$-wall, that is from ${\mathcal{O}}_{0}$ to ${\mathcal{O}}_{\lambda}$.

For this we recall the translation functor

[TABLE]

which is defined as the unique (up to isomorphism) indecomposable projective functor from ${\mathcal{O}}_{0}$ to ${\mathcal{O}}_{\lambda}$ that maps ${\Delta}(0)$ to ${\Delta}(\lambda)$, see [BG80]. The functor $T_{0}^{\lambda}$ is exact, has both adjoints, and commutes with simple preserving duality on $\mathcal{O}$. It acts on Verma modules and simple modules in the following way, see [Ja79, Chapter 2]:

Proposition 10**.**

For $w\in W$, we have:

• •

$T_{0}^{\lambda}{\Delta}(w\cdot 0)={\Delta}(w\cdot\lambda)$,

• •

$T_{0}^{\lambda}L(w\cdot 0)=\begin{cases}L(w\cdot\lambda),&\text{if }w\in\tilde{W}^{\lambda};\\ 0,&\text{otherwise}.\end{cases}$**

Therefore, a homomorphism ${\Delta}(x^{\prime}\cdot 0)\hookrightarrow{\Delta}(x\cdot 0)$ in $\boldsymbol{\Delta}$ can be mapped either to an isomorphism, in case $x^{\prime}$ and $x$ are in the same $W_{\lambda}$-coset, or, otherwise, to a non-surjective monomorphism ${\Delta}(x^{\prime}\cdot\lambda)\hookrightarrow{\Delta}(x\cdot\lambda)$. An isomorphism between Verma modules is necessarily a non-zero scalar times the identity. So, for all $x\in\tilde{W}^{\lambda}$ with $x\geq w$, all the maps in the part of $\boldsymbol{\Delta}$ indexed by $[w,w_{0}]\cap xW_{\lambda}$ are mapped to non-zero multiples of the identity. We will write equalities for such maps in the diagrammatic presentation of $T_{0}^{\lambda}(\boldsymbol{\Delta})$.

Lemma 11**.**

For any $x\in[w,w_{0}]$, the module ${\Delta}(x\cdot\lambda)$ is neither a source, nor a sink, of an equality in the complex $T_{0}^{\lambda}(\boldsymbol{\Delta})$ if and only if $x\in X_{w}$.

Proof.

Fix $x\in[w,w_{0}]$, and denote by $\Delta_{x}$ the piece of $\boldsymbol{\Delta}$ indexed by $[w,w_{0}]\cap xW_{\lambda}$. All homomorphisms in $\Delta_{x}$ are mapped to isomorphisms in $T_{0}^{\lambda}(\Delta_{x})$, and this piece is connected as a graph by Lemma 5. Moreover, there are no isomorphisms in $T_{0}^{\lambda}(\boldsymbol{\Delta})$ that enter or exit $T_{0}^{\lambda}(\Delta_{x})$. The claim now follows from Lemma 8 and the definition of $X_{w}$. ∎

Example 12**.**

In type $A_{3}$, take $\lambda+\rho=(2,1,1,0)$. Then $S=\{\alpha_{2}\}$. Take $w=s_{1}s_{2}$. The complex (3) in the regular block is displayed in Figure 1.

Arrows show the directions of homomorphisms (note that they have the opposite directions compared to the arrows in $W$). After applying $T_{0}^{\lambda}$, we get the complex in ${\mathcal{O}}_{\lambda}$ displayed in Figure 2. Elements of $\tilde{W}^{\lambda}$ are displayed in bold font, and those in $X_{w}$ are underlined.

4.3. Cutting off the equalities

The second step is to cut off the equalities that appear in $T_{0}^{\lambda}(\boldsymbol{\Delta})$, so that only the part indexed by $X_{w}$ remains.

Lemma 13**.**

If a sequence of modules

[TABLE]

is a complex (resp. exact), then so is

[TABLE]

Proof.

A direct computation. ∎

Fix $x\in\tilde{W}^{\lambda}\setminus X_{w}$ with $x\geq w$. We partition the intersection $[w,w_{0}]\cap xW_{\lambda}$ into a disjoint union of $2$-subsets as in Lemma 6. Denote such a partition by ${\mathcal{P}}_{x}$. We want to apply inductively Lemma 13 where \textstyle{G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{G}

corresponds to a $2$-subset in ${\mathcal{P}}_{x}$, so that we eventually exhaust all of $[w,w_{0}]\cap xW_{\lambda}$. Then we repeat the procedure for all such $x$. However, we have to check that, at each step, the differential of our new complex has not changed in an essential way.

Lemma 14**.**

Choose $x\in\tilde{W}^{\lambda}\setminus X_{w}$ with $x\geq w$, and $\{x_{1}\to x_{2}\}\in{\mathcal{P}}_{x}$. The following diagrams cannot appear as subdiagrams in $T_{0}^{\lambda}(\boldsymbol{\Delta})$:

1. $($a$)$

for $y,z\in X_{w}$ such that $y\to z$:

[TABLE] 2. $($b$)$

for $y\in\tilde{W}^{\lambda}$, $y\neq x$, $y_{1},y_{2}\in[w,w_{0}]\cap yW_{\lambda}$ such that $y_{1}\to y_{2}$:

[TABLE] 3. $($c$)$

for $\{x_{1}\to x_{2}\},\{x_{3}\to x_{4}\}\in{\mathcal{P}}_{x}$ with $l(x_{1})=l(x_{3})$ and $x_{1}\neq x_{3}$:

[TABLE]

Proof.

Consider diagram (a). If there is a non-zero morphism ${\Delta}(z\cdot\lambda)\to{\Delta}(x_{1}\cdot\lambda)$, then $x_{1}\leq z$. By [BB06, Proposition 2.5.1], we have $x\leq z$. But then $\mu^{\lambda}(w,x)=0$ implies $\mu^{\lambda}(w,z)=0$, which is a contradiction with $z\in X_{w}$.

In diagram (b), the diagonal arrows are injections, but not surjections. The composition ${\Delta}(y_{2}\cdot\lambda)\to{\Delta}(x_{1}\cdot\lambda)={\Delta}(x_{2}\cdot\lambda)\to{\Delta}(y_{1}\cdot\lambda)$ would map the highest weight space of ${\Delta}(y_{2}\cdot\lambda)$ into the radical of ${\Delta}(y_{1}\cdot\lambda)$. But this is impossible since $y_{1}\cdot\lambda=y_{2}\cdot\lambda$.

The fact that diagram (c) is not possible follows from the choice of the partition ${\mathcal{P}}_{x}$, namely, Lemma 6(b). ∎

Lemma 14 has the following consequence:

Corollary 15**.**

When using Lemma 13 to cut off an equality ${\Delta}(x_{2}\cdot\lambda)={\Delta}(x_{1}\cdot\lambda)$ corresponding to $\{x_{1}\to x_{2}\}\in{\mathcal{P}}_{x}$ from $T_{0}^{\lambda}(\boldsymbol{\Delta})$, the following holds:

1. $($a$)$

No morphism is changed between Verma modules indexed by two elements from $X_{w}$, 2. $($b$)$

No isomorphism is changed between Verma modules indexed by elements from a different coset $yW_{\lambda}$, 3. $($c$)$

No isomorphism is changed corresponding to some other pair in the same partition ${\mathcal{P}}_{x}$.

Therefore, we can cut-off pairs from ${\mathcal{P}}_{x}$, for all $x\in\tilde{W}^{\lambda}\setminus X_{w}$ with $x\geq w$, in any order. This implies the following statement.

Theorem 16**.**

Let $\lambda$ be dominant and integral and $w\in\tilde{W}^{\lambda}$.

1. $($a$)$

One can choose non-zero monomorphisms as explained in Subsection 4.1 so that the singular BGG complex (2) becomes a complex. 2. $($b$)$

If the regular BGG complex (3) of $L(w\cdot 0)$ is exact, then so is the singular BGG complex (2) of $L(w\cdot\lambda)$.

The converse of the second statement above is not true in general, since the translation functor can kill homology of the regular BGG complex, if it consists of simple modules not parameterized by the maximal coset representatives (see Proposition 10). The smallest example is the following.

Example 17**.**

In type $B_{3}$ take $\lambda+\rho=(1,0,0)$. Then $S=\{\alpha_{2},\alpha_{3}\}$. Take

[TABLE]

We want to show that the regular BGG complex of $L(w\cdot 0)$ is not exact, but that the singular one corresponding to $L(w\cdot\lambda)$ is exact.

Denote by $A$ the set of immediate ascendants of $w$, i.e., $A:=\{x\in W\colon w\to x\}$. Denote also:

[TABLE]

One can check that the right hand side is a reduced expression of $v$, that $v\notin\tilde{W}^{\lambda}$, and that $w<v$. One can calculate the following Kazhdan-Lusztig polynomials:

[TABLE]

From [Ir90, Theorem 1.4.1] it follows that the first radical layer ${\operatorname{rad}}^{1}{\Delta}(w\cdot 0)$ consists of the composition factors $L(x\cdot 0)$ for $x\in A\cup\{v\}$. Consider the first part of the regular BGG complex (3):

[TABLE]

Clearly ${\operatorname{Ker}}(d_{0})={\operatorname{rad}}\,{\Delta}(w\cdot 0)$. The Verma modules on the left hand side map onto the submodules of ${\Delta}(w\cdot 0)$ with heads $L(x\cdot 0)\subseteq{\operatorname{rad}}^{1}{\Delta}(w\cdot 0)$ with $x\in A$. So $L(v\cdot 0)\subseteq{\operatorname{rad}}^{1}{\Delta}(w\cdot 0)$ is not in the image of $d_{1}$, and hence the zero-th homology group of the regular BGG complex is non-zero since ${\operatorname{Ker}}(d_{0})/\mathrm{Im}(d_{1})\twoheadrightarrow L(v\cdot 0)$. In fact, from (4), we may deduce ${\operatorname{Ker}}(d_{0})/\mathrm{Im}(d_{1})\cong L(v\cdot 0)$.

On the other hand, one can check that $\tilde{W}^{\lambda}$ is a linear poset:

[TABLE]

We want to show that $X_{w}=\{w\to s_{1}w\}$. For this, we need to find $z\not\in\tilde{W}^{\lambda}$ such that $w<x<s_{2}s_{1}w$. One can check that $z=s_{2}s_{3}s_{1}s_{2}s_{3}$ does the job. So, we see that our singular BGG complex is:

[TABLE]

From (4) and [Ir90, Theorem 1.4.2] it follows that ${\operatorname{rad}}^{1}{\Delta}(w\cdot\lambda)$ is just $L(s_{1}w\cdot\lambda)$. This implies that (5) is exact.

In the next section we give a computable criteria for a singular BGG complex to be exact.

The singular BGG complex of a dominant simple module admits a simpler combinatorial description. Choose a dominant weight $\lambda^{c}$ such that $\lambda^{c}+\rho$ is orthogonal precisely to those simple roots $\alpha$ which are not orthogonal to $\lambda+\rho$. The subgroup $W_{\lambda^{c}}$ is complementary to $W_{\lambda}$, it is generated by all simple non-singular reflections.

Proposition 18**.**

We have $X_{w_{0}^{\lambda}}=W_{\lambda^{c}}\cdot w_{0}^{\lambda}$. In other words, one can identify weights which appear in the singular BGG complex of $L(w_{0}^{\lambda}\cdot\lambda)=L(\lambda)$ with the orbit of $\lambda$ under the complementary parabolic subgroup.

Proof.

Denote by $Y$ the subset of $W^{\lambda}$ consisting of those $w$ for which $[e,w]\subseteq W^{\lambda}$. It is clear that $Y\cdot w_{0}^{\lambda}=X_{w_{0}^{\lambda}}$, and thus we need to prove that $Y=W_{\lambda^{c}}$. The claim that $W_{\lambda^{c}}\subseteq Y$ follows from the subword property. For the converse, suppose that $w\in W^{\lambda}\setminus W_{\lambda^{c}}$. Then, a fixed reduced expression of $w$ must contain some singular reflection $s$. But then $s<w$ and $s\notin W^{\lambda}$ and hence $w\notin Y$. ∎

The description of the parameter set of the singular BGG complex with a dominant highest weight given by Proposition 18 is related to [FM98]. See Section 9.

5. Exactness and KLV-polynomials

5.1. Singular KLV polynomials

In [Ir90], the following singular version of the KLV (Kazhdan-Lusztig-Vogan) polynomials was introduced. Take an integral, possibly singular weight $\mu$ which is antidominant, i.e., $w_{0}\cdot\mu$ is dominant, and for $y,z$ in $W^{\mu}$ put

[TABLE]

It is a polynomial, equal to [math] unless $y\leq z$, equal to $1$ if $y=z$, and, in general, of degree at most $(l(z)-l(y)-1)/2$.

If $\mu$ is regular, then $P^{\mu}_{y,z}=P_{y,z}$ agrees with the usual KL (Kazhdan-Lusztig) polynomials, by a result of Vogan. In general, in [So89, Ir90] the following formula that relates the singular KLV-polynomials to the usual ones is proved:

Proposition 19**.**

For $\mu,y,z$ as above, we have

[TABLE]

Let us return to our dominant parameterization by $\lambda$. If we put $\mu=w_{0}\cdot\lambda$, then it is easy to see that $W_{\mu}=w_{0}W_{\lambda}w_{0}$, and $W^{\mu}w_{0}=\tilde{W}^{\lambda}$. For $w,x\in\tilde{W}^{\lambda}$ with $w\leq x$ the relevant KLV-polynomial is

[TABLE]

Theorem 20**.**

For $w\in\tilde{W}^{\lambda}$, the following statements are equivalent:

1. $($a$)$

The singular BGG complex (2) of $L(w\cdot\lambda)$ is exact. 2. $($b$)$

For all $i\geq 0$, we have

[TABLE]

where ${\mathbb{C}}_{x\cdot\lambda}$ is the $1$-dimensional ${\mathfrak{h}}$-module with weight $x\cdot\lambda$. 3. $($c$)$

For all $i\geq 0$ and $x\in\tilde{W}^{\lambda}$, we have

[TABLE] 4. $($d$)$

For all $x\in\tilde{W}^{\lambda}$ with $x\geq w$, we have $P^{w_{0}\cdot\lambda}_{xw_{0},ww_{0}}(q)=|\mu^{\lambda}(w,x)|$.

Proof.

Having Theorem 16, the proof now follows [BH09, 3.4. and 4.3.]. We outline it here, for the sake of completeness.

Assume claim (a). The Killing form induces an ${\mathfrak{h}}$-isomorphism

[TABLE]

and the latter is the $i$-th right derived functor of ${\mathbb{C}}\otimes_{U({\mathfrak{n}}^{-})}-$. It can be easily calculated on $L(w\cdot\lambda)$ by using (2) as a free $U({\mathfrak{n}}^{-})$ resolution, and claim (b) follows.

Assume claim (b). Then $L(w\cdot\lambda)$ is a generalized Kostant module in the sense of [EH04], and their Theorem 2.8. shows that (2) must be exact, proving claim (a).

Claims (b) and (c) are equivalent by [Sc81, Lemma 5.13], where it is shown that any $\nu$-weight space of $H^{i}\left({\mathfrak{n}}^{+},L(w\cdot\lambda)\right)$ is, in fact, isomorphic to ${\operatorname{Ext}}^{i}_{\mathcal{O}}\left({\Delta}(\nu),L(w\cdot\lambda)\right)$.

Claims (c) and (d) are equivalent by (6) and the definition of $X^{i}_{w}$. ∎

5.2. Kostant modules

In [BH09], it is shown that the BGG complex of a simple module in the parabolic category ${\mathcal{O}}^{\lambda}_{0}$ is exact precisely when the simple module is a Kostant module. It turns out that this is not true in ${\mathcal{O}}_{\lambda}$; a simple module in ${\mathcal{O}}_{\lambda}$ whose BGG complex is exact does not have to be a Kostant module in the sense of [BH09, Subsection 3.3]. The problem is that our $X_{w}$ does not have to be an interval in $\tilde{W}^{\lambda}$. The smallest example can be found in type $A_{3}$. Let $\lambda+\rho=(2,1,1,0)$, then $S=\{\alpha_{2}\}$ and take $w=s_{3}s_{1}s_{2}$. The posets $[w,w_{0}]\supseteq[w,w_{0}]\cap\tilde{W}^{\lambda}\supseteq X_{w}$ are shown in Figure 3. Elements of $\tilde{W}^{\lambda}$ are displayed in bold font, those in $X_{w}$ are underlined, and the arrows follow the Bruhat order.

The phenomenon is related to the fact that certain generalized Verma module in the regular parabolic block ${\mathcal{O}}^{\lambda}$ does not have simple socle. This example suggests that their definition of Kostant modules might not be the “correct one” for singular blocks. We will return to this question later in Subsection 6.4.

6. Koszul duals of modules with the BGG resolution

6.1. Parabolic category ${\mathcal{O}}^{\lambda}$

A dominant, integral, possibly singular weight $\lambda$ uniquely determines a parabolic subgroup ${\mathfrak{p}}\subseteq{\mathfrak{g}}$ containing ${\mathfrak{b}}$, which, in turn, defines the parabolic category ${\mathcal{O}}^{\lambda}$, that is, the full subcategory of ${\mathcal{O}}$ consisting of ${\mathfrak{p}}$-locally finite modules. Note that ${\mathcal{O}}^{0}={\mathcal{O}}$. We denote by ${\mathcal{O}}^{\lambda}_{\nu}$ the block of ${\mathcal{O}}^{\lambda}$ corresponding to the central character determined by the weight $\nu$. In particular, consider the regular block ${\mathcal{O}}^{\lambda}_{0}$. Generalized Verma modules ${\Delta}^{\lambda}(w\cdot 0)$, and simple modules $L(w\cdot 0)$ in ${\mathcal{O}}^{\lambda}_{0}$ are parameterized by $w$ running over the set ${}^{\lambda}W$ of the minimal representatives of the cosets $W_{\lambda}\backslash W$. It is easy to see that the map $w\mapsto w^{-1}w_{0}$ gives a bijection $\tilde{W}^{\lambda}\stackrel{{\scriptstyle\sim}}{{\longrightarrow}}{}^{\lambda}W$. Denote

[TABLE]

Remark 21**.**

The map $w\mapsto\hat{\hat{w}}=w_{0}ww_{0}$ gives rise to an automorphism of the Dynkin diagram, and hence of the Coxeter system. It is equal to the identity in all simple Coxeter types except $A_{n}$ and odd rank $D_{n}$, when it is the unique non-identity automorphism of the Dynkin diagram. In particular, the map preserves for KL- and KLV-polynomials, multiplicities and ${\operatorname{Ext}}$-groups from generalized Verma modules to simple modules.

6.2. Graded version of category ${\mathcal{O}}$

In what follows, by graded we mean $\mathbb{Z}$-graded.

For a block ${\mathcal{O}}^{\lambda}_{\nu}$, denote by $E^{\lambda}_{\nu}$ the endomorphism algebra of a minimal projective generator in ${\mathcal{O}}^{\lambda}_{\nu}$. The category ${\mathcal{O}}^{\lambda}_{\nu}$ is equivalent to $E^{\lambda}_{\nu}\text{-}\mathrm{mod}$, the category of finitely generated $E^{\lambda}_{\nu}$-modules. In [BGS96], it is proved that $E^{0}_{\lambda}$ and $E^{\lambda}_{0}$ have a Koszul grading, and, moreover, that they are Koszul dual to each other. This was generalized to arbitrary ${\mathcal{O}}^{\lambda}_{\nu}$ in [Ba99], see also [Ma09a].

For a graded algebra $E$, denote denote by $E$-${\operatorname{gmod}}$ the category of graded finitely generated $E$-modules, where for morphisms we take only homogeneous homomorphisms of degree zero. All structural modules, that is simple modules, (generalized) Verma modules and their duals, indecomposable projectives, injectives and tilting modules in both ${\mathcal{O}}^{0}_{\lambda}$ and ${\mathcal{O}}^{\lambda}_{0}$ admit a graded lift to $E^{0}_{\lambda}$-${\operatorname{gmod}}$ and $E^{\lambda}_{0}$-${\operatorname{gmod}}$, respectively, and this lift is unique up to a shift in grading. For modules with simple head, the grading is given by their radical filtration. In particular, this applies to simple modules, generalized Verma modules, and indecomposable projectives. We will use the same notation for their graded lifts, assuming that their heads have degree [math]. In the latter case, their radicals are contained in positive degrees.

For a graded module $\displaystyle M=\bigoplus_{i\in{\mathbb{Z}}}M_{i}$, we denote by $M\langle k\rangle$ the same module with the shifted grading: $M\langle k\rangle_{i}=M_{i+k}$. Our BGG complex (2) admits a graded lift, that is it can be lifted to $E^{0}_{\lambda}$-${\operatorname{gmod}}$, and, moreover, this lift is linear in the sense that the $i$-th term of the complex is a direct sum of ${\Delta}(x\cdot\lambda)\langle i\rangle$, where $x$ varies. The analogous statements are true for BGG complexes in regular parabolic blocks.

Remark 22**.**

It can be shown that any minimal resolution of a simple module in ${\mathcal{O}}_{\lambda}$ by direct sums of generalized Verma modules lifts to a linear resolution in $E^{0}_{\lambda}$-${\operatorname{gmod}}$. This is true as ${\operatorname{Hom}}$-spaces between Verma modules are $1$-dimensional and gradable.

For our further discussion we will use the following statement which is [BGS96, Proposition 1.3.1]:

Proposition 23**.**

For $w,x\in\tilde{W}^{\lambda}$ and for all $i\geq 0$, we have:

1. $($a$)$

$\dim{\operatorname{Ext}}_{\mathcal{O}}^{i}\left({\Delta}(x\cdot\lambda),L(w\cdot\lambda)\right)=\left[{\operatorname{rad}}^{i}{\Delta}^{\lambda}(\hat{x}\cdot 0)\colon L(\hat{w}\cdot 0)\right]$, 2. $($b$)$

$\dim{\operatorname{Ext}}_{\mathcal{O}^{\lambda}}^{i}\left({\Delta}^{\lambda}(\hat{x}\cdot 0),L(\hat{w}\cdot 0)\right)=\left[{\operatorname{rad}}^{i}{\Delta}(x\cdot\lambda)\colon L(w\cdot\lambda)\right]$,

where ${\operatorname{rad}}^{i}$ means the $i$-th layer of the radical filtration.

We will also need the following graded version of the BGG reciprocity (which is proved by the same argument as the classical result):

Proposition 24** (Graded BGG reciprocity).**

For all weights $\nu,\xi$ and all $i\geq 0$, we have

[TABLE]

An analogous statement holds in ${\mathcal{O}}^{\lambda}$.

6.3. (Generalized-)Verma flags of projectives

By Koszul duality, a simple module $L(w\cdot\lambda)\in{\mathcal{O}}_{\lambda}$, where $w\in\tilde{W}^{\lambda}$, corresponds to the indecomposable projective module $P^{\lambda}(\hat{w}\cdot 0)\in{\mathcal{O}}^{\lambda}_{0}$, where $\hat{w}=w^{-1}w_{0}\in{}^{\lambda}W$. A simple module $L(\hat{w}\cdot 0)\in{\mathcal{O}}^{\lambda}_{0}$ corresponds to the indecomposable projective module $P(w\cdot 0)\in{\mathcal{O}}_{\lambda}$. We note that, in the classical Koszul duality, a simple module corresponds to an indecomposable injective module in the Koszul dual picture. Therefore, we need to compose the classical Koszul duality with the usual simple preserving duality on ${\mathcal{O}}$ to get a projective module. This is related to Remark 21.

We want to see how the exactness of the BGG complex of a simple module in one category is reflected on its Koszul dual projective object.

Proposition 25**.**

For $w\in\tilde{W}^{\lambda}$, the following statements are equivalent:

1. $($a$)$

The BGG complex of $L(\hat{w}\cdot 0)\in{\mathcal{O}}^{\lambda}_{0}$ is exact. 2. $($b$)$

The projective $P(w\cdot\lambda)\in{\mathcal{O}}_{\lambda}$ has a multiplicity-free Verma flag. 3. $($c$)$

$\left[{\Delta}(\lambda)\colon L(w\cdot\lambda)\right]=1$. 4. $($d$)$

${\operatorname{End}}(P(w\cdot\lambda))$* is commutative.*

Proof.

From [BH09, Subsection 3.4], Proposition 23(b) and the BGG reciprocity, it follows that claim (a) is equivalent to the following statement: For all $x\in\tilde{W}^{\lambda}$, $x\leq w$, and $i\geq 0$, we have:

[TABLE]

Obviously this implies claim (b).

Assume claim (b), and take $x\in\tilde{W}^{\lambda}$, $x\leq w$ and $i\geq 0$. Recall that $L(w\cdot\lambda)$ must always appear in ${\operatorname{rad}}^{l(w)-l(x)}{\Delta}(x\cdot\lambda)$ exactly once (this follows, for example, from [Ir90, Section 1.4.2] and the fact that non-zero KL-polynomials always have $1$ as the constant term). From this, it follows that, if $i=l(w)-l(x)$, then $\left(P(w\cdot\lambda)\colon{\Delta}(x\cdot\lambda)\langle i\rangle\right)=1$.

So, assume $i\neq l(w)-l(x)$ and $\left(P(w\cdot\lambda)\colon{\Delta}(x\cdot\lambda)\langle i\rangle\right)>0$. From the above, we know that $\left(P(w\cdot\lambda)\colon{\Delta}(x\cdot\lambda)\langle l(w)-l(x)\rangle\right)=1$. This implies that ${\Delta}(x\cdot\lambda)$ occurs more than once in $P(w\cdot\lambda)$, which is a contradiction. This establishes (7), and hence implies claim (a).

Claims (b) and (c) are equivalent by BGG reciprocity. Claim (d) is equivalent to claim (c) by [St04, Theorem 7.1]. ∎

A similar result in the other direction is weaker, due to the fact that projective modules in ${\mathcal{O}}^{\lambda}_{0}$ are much more complicated than projective modules in ${\mathcal{O}}_{\lambda}$, as the following example suggests.

Example 26**.**

In type $A_{3}$, take $\lambda+\rho=(1,1,0,0)$ and $w=s_{3}s_{1}$. Then $S=\{s_{1},s_{3}\}$. The BGG complex of $L(w\cdot\lambda)\in{\mathcal{O}}_{\lambda}$ is not exact. However, its Koszul-dual is the projective module $P^{\lambda}(s_{2}s_{3}s_{1}s_{2}\cdot 0)\in{\mathcal{O}}^{\lambda}_{0}$, which has a multiplicity-free generalized Verma flag consisting of:

[TABLE]

Note that the components ${\Delta}^{\lambda}(s_{2}s_{3}s_{1}s_{2}\cdot 0)$ and ${\Delta}^{\lambda}(s_{2}s_{3}s_{1}\cdot 0)\langle 1\rangle$ are “predictable” in the sense that they correspond to trivial KL-polynomials, while the other two, ${\Delta}^{\lambda}(s_{2}\cdot 0)\langle 1\rangle$ and ${\Delta}^{\lambda}(e\cdot 0)\langle 2\rangle$, are coming from non-trivial (however, monomial) KL-polynomials.

Recall that a generalized Verma module ${\Delta}^{\lambda}(\hat{x}\cdot 0)$, where $\hat{x}\in{}^{\lambda}W$, is the maximal quotient of ${\Delta}(\hat{x}\cdot 0)\in{\mathcal{O}}$ that belongs to the category ${\mathcal{O}}^{\lambda}$. The kernel of this quotient consists of all submodules ${\Delta}(\hat{y}\cdot 0)\subset{\Delta}(\hat{x}\cdot 0)$ for which $\hat{y}\notin{}^{\lambda}W$, see [Hu08, Page 187]. From this it follows that the head of a Verma module ${\Delta}(\hat{w}\cdot 0)$, where $\hat{w}\in{}^{\lambda}W$, survives in ${\operatorname{rad}}^{l(\hat{w})-l(\hat{x})}{\Delta}^{\lambda}(\hat{x}\cdot 0)$ if and only if there is no $\hat{y}$ such that $\hat{x}\leq\hat{y}\leq\hat{w}$ and $\hat{y}\notin{}^{\lambda}W$. This is equivalent to $\mu^{\lambda}(w,x)\neq 0$, that is, to $x\in X_{w}$. We say that this particular occurrence of $L(\hat{w}\cdot 0)$ in $\Delta^{\lambda}(\hat{x}\cdot 0)$ is predictable. In this case, by the BGG reciprocity, we also have

[TABLE]

This particular occurrence of ${\Delta}^{\lambda}(\hat{x}\cdot 0)$ in $P^{\lambda}(\hat{w}\cdot 0)$ is also said to be predictable. In conclusion, we say that an occurrence of either a simple in a Verma, or a Verma in a projective to be predictable, if the shift in grading is precisely given by the difference of lengths of the parameters. Note that, in ${\mathcal{O}}_{\lambda}$, an occurrence is predictable if and only if the corresponding (ungraded) multiplicity is one.

Remark 27**.**

Note the following instance of the Koszul duality. In ${\mathcal{O}}_{\lambda}$, BGG complexes are given by $X_{w}\subseteq\tilde{W}^{\lambda}$ which also coincides with the support of the function $\mu^{\lambda}(w,-)$, and the predictable factors are given by the interval $[w,w_{0}]\cap\tilde{W}^{\lambda}$. In ${\mathcal{O}}^{\lambda}_{0}$, the roles are reversed (up to the bijection $x\mapsto\hat{x}$).

Proposition 28**.**

For $w\in\tilde{W}^{\lambda}$, the following statements are equivalent:

1. $($a$)$

The singular BGG complex (2) of $L(w\cdot\lambda)\in{\mathcal{O}}_{\lambda}$ is exact. 2. $($b$)$

The module $P^{\lambda}(\hat{w}\cdot 0)\in{\mathcal{O}}^{\lambda}_{0}$ has a generalized Verma flag consisting only of predictable factors.

Note that a generalized Verma flag consisting only of predictable factors is necessarily a multiplicity free flag.

Proof.

From Theorem 20(c), Proposition 23(a), and the BGG reciprocity it follows that claim (a) is equivalent to the following statement: For all $x\in\tilde{W}^{\lambda}$, $x\geq w$ and $i\geq 0$, we have:

[TABLE]

This is precisely claim (b). ∎

Now we can show the necessity of using $X_{w}$ as the indexing set for our singular BGG complexes.

Proposition 29**.**

Suppose we have a resolution in ${\mathcal{O}}_{\lambda}$ of the form

[TABLE]

for some multiplicities $m(x)>0$, which lifts to a linear resolution in $E^{0}_{\lambda}$-${\operatorname{gmod}}$. Then all $m(x)=1$ and $Y^{i}=X^{i}_{w}$, for all $i\geq 0$.

Proof.

From linearity and exactness, it follows that, for every $x\in Y^{i}$, we must have $x\geq w$ and $l(x)=l(w)+i$. From this it follows that each ${\Delta}(x\cdot\lambda)$, where $x\in Y^{i}$, lifts to ${\Delta}(x\cdot\lambda)\langle l(x)-l(w)\rangle\in E^{0}_{\lambda}$-${\operatorname{gmod}}$.

Consider the Euler characteristic of the lift of the above resolution in the graded Grothendieck group:

[TABLE]

Here we put $m(x)=0$ if $\displaystyle x\not\in\bigcup_{i}Y_{i}$.

Fix $x\in[w,w_{0}]\cap\tilde{W}^{\lambda}$. For any $z\in[w,x]\cap\tilde{W}^{\lambda}$, the Verma module ${\Delta}(z\cdot\lambda)$ contains the predictable occurrence of $L(x\cdot\lambda)\langle l(x)-l(z)\rangle$. This occurrence appears in (8) as $L(x\cdot\lambda)\langle l(x)-l(w)\rangle$, independent of $z$. A potential non-predictable occurrence can appear only in a strictly lower graded component of (8). It follows that all the predictable occurrences must cancel out, i.e.,

[TABLE]

By the Möbius inversion formula, see e.g. [Sta11, Proposition 3.7.1.], we have

[TABLE]

This completes the proof. ∎

Proposition 29 also follows from Theorem 33, but the proof presented here is much more elementary.

6.4. Kostant modules revisited

We say that a module $L(w\cdot\lambda)$, where $w\in\tilde{W}^{\lambda}$, is a Kostant module, if there is a convex subset (which is not necessarily an interval) $Y\subseteq[w,w_{0}]\cap\tilde{W}^{\lambda}$ such that, for all $i\geq 0$, we have

[TABLE]

Proposition 30**.**

For $w\in\tilde{W}^{\lambda}$, the following statements are equivalent:

1. $($a$)$

The singular BGG complex (2) of $L(w\cdot\lambda)\in{\mathcal{O}}_{\lambda}$ is exact. 2. $($b$)$

$L(w\cdot\lambda)$* is a Kostant module in the above sense.*

By Proposition 30, the two notions of “Kostant module” agree on the regular blocks of category ${\mathcal{O}}$.

Proof.

Given claim (a), Theorem 20(b) implies claim (b) with $Y=X_{w}$. ’

Assume claim (b), for some $Y$, and recall that

[TABLE]

and that the right hand side has dimension equal to $\left(P^{\lambda}(\hat{w}\cdot 0)\colon{\Delta}^{\lambda}(\hat{x}\cdot 0)\langle i\rangle\right)$. Since $i=l(x)-l(w)$, the dimension is positive if and only if the occurrence of ${\Delta}^{\lambda}(\hat{x}\cdot 0)\langle i\rangle$ in $P^{\lambda}(\hat{w}\cdot 0)$ is predictable. Therefore, we obtain $Y=X_{w}$ implying claim (a). ∎

For general singular-parabolic blocks, the convexness of $Y$ in the definition of Kostant modules should probably be expressed in terms of the “$\mu$-order” as suggested in [BH09, Subsection 9.3], instead of the Bruhat order. These two orders agree on ${\mathcal{O}}_{\lambda}$ and on ${\mathcal{O}}^{\lambda}_{0}$.

7. Non-Kostant modules in low ranks

In this section, we use Theorem 20(d) together with Proposition 19 to list non-Kostant modules, i.e., simple modules whose singular BGG complex is not exact, in all singular blocks of category ${\mathcal{O}}$ in ranks up to $4$ for classical Lie algebras, and for one large singularity in the exceptional case $F_{4}$.

In the first column of the following tables, we put singularity sets $S$ which determine the blocks ${\mathcal{O}}_{\lambda}$ (where $\lambda$ is a dominant integral weight with singularity $S$). In the second column we list all $w\in\tilde{W}^{\lambda}$ such that $L(w\cdot\lambda)\in{\mathcal{O}}_{\lambda}$ is not Kostant.

Instead of $S=\{\alpha_{i_{1}},\ldots,\alpha_{i_{j}}\}$ we will just write $S=\{i_{1},\ldots,i_{j}\}$. Similarly, instead of $w=s_{i_{1}}\ldots s_{i_{j}}$ we will write $w=(i_{1}\ldots i_{j})$. All expressions will be reduced.

The calculations were performed in SageMath, version 8.4.

7.1. Rank 1 and 2

There are no non-Kostant modules in ranks $1$ and $2$. This follows from the fact that all BGG complexes are exact already in the regular block, which is a consequence of the fact that all $KL$-polynomials are trivial.

7.2. Rank 3

In type $A_{3}$, we only present possible singularity sets up the unique non-identity automorphism of the Dynkin diagram which swaps $1\leftrightarrow 3$. In type $B_{3}=C_{3}$ we list all walls with non-Kostant modules.

[TABLE]

[TABLE]

7.3. Rank 4

As before, in type $A_{4}$, all other singular walls can be obtained by by exchanging $1\leftrightarrow 4$ and $2\leftrightarrow 3$. In type $D_{4}$, all other singular walls can be obtained permuting $1$, $3$ and $4$.

Again, for type $B_{4}$, all the walls containing non-Kostant modules are given.

[TABLE]

[TABLE]

[TABLE]

Note that most of the BGG complexes resolving a module of a dominant singular highest weight are not exact, in contrast to the regular case where all the BGG complexes of modules of a dominant highest weight are, in fact, exact.

For the exceptional case $F_{4}$, due to the computational complexity, we present only one type of singularity, which has the least number of maximal representatives. However, the full exposition of all the blocks would probably take up several pages.

[TABLE]

8. BGG complexes for balanced quasi-hereditary algebras

8.1. Balanced quasi-hereditary algebras

In this section we work in the setup of [Ma09b]. Let $A$ be a finite dimensional positively graded quasi-hereditary algebra over an algebraically closed field. Denote by $\{e_{\lambda}\colon\lambda\in\Lambda\}$ a complete set of pairwise-orthogonal primitive idempotents of $A$ with a fixed linear order on $\Lambda$ that defines the quasi-hereditary structure. Denote by $A$-${\operatorname{gmod}}$ the category of all finite-dimensional graded $A$-modules, where morphisms are homogeneous homomorphisms of degree zero. This is an abelian category with enough projectives and enough injectives. Denote by $\langle\cdot\rangle$ the shift in grading: $M\langle i\rangle_{j}:=M_{i+j}$. Denote by $L(\lambda)$, $\Delta(\lambda)$, $\nabla(\lambda)$, $P(\lambda)$, $I(\lambda)$, and $T(\lambda)$ respectively the simple, standard, costandard, projective, injective, and tilting module corresponding to $\lambda\in\Lambda$. We fix their graded shifts so that $L(\lambda)$, $\Delta(\lambda)$ and $P(\lambda)$ have top in degree zero, $\nabla(\lambda)$ and $I(\lambda)$ have socle in degree zero, and $T(\lambda)$ has in degree zero the unique subquotient isomorphic to $L(\lambda)$. These modules are called the structural modules. If $M$ is a structural module, we will say that $M\langle i\rangle$ is centered at $-i$.

Denote by $\mathcal{D}^{b}(A)$ the bounded derived category of $A$-${\operatorname{gmod}}$. A complex

[TABLE]

of direct sums of structural modules of the same kind is said to be linear, provided that all indecomposable direct summands of each $X^{i}$ are centered at $i$. Denote by $[i]$ the homological shift normalized such that $X[i]^{j}=X^{i+j}$.

Assume that $A$ is balanced in the sense of [Ma09b]. This means that each standard module $\Delta(\lambda)$ has a linear tilting coresolution $0\to\Delta(\lambda)\to{\mathcal{S}}_{\lambda}^{\bullet}$, and each costandard module $\nabla(\lambda)$ has a linear tilting resolution $\mathcal{C}_{\lambda}^{\bullet}\to\Delta(\lambda)\to 0$. The main examples for our purposes are graded versions of blocks of the category ${\mathcal{O}}$, see Subsection 6.2. That these are given by balanced algebras is proved in [Ma09a, Proposition 2.7] for the regular block, and [Ma09a, Section 4] for singular blocks.

In what follows we will use the following statement, cf. [Ma09b, Corollary 7 and Proposition 5]:

Proposition 31**.**

For a balanced quasi-hereditary algebra $A$, the following holds:

1. $($a$)$

$A$* is Koszul, i.e. the minimal projective resolution of each $L(\lambda)$ is linear.* 2. $($b$)$

Each $L(\lambda)$ is isomorphic in $\mathcal{D}^{b}(A)$ to a linear complex of tilting modules. In other words, there is a linear complex $\mathcal{T}_{\lambda}^{\bullet}$ of tilting modules such that

[TABLE]

Let us mention that a balanced algebra is also necessarily standard Koszul in the sense of [ADL03], which means that standard modules have linear projective resolutions, and costandard modules have linear injective coresolutions. This is stronger than just Koszulity.

8.2. BGG complexes for balanced algebras

In this setup, by a BGG complex of $L(\lambda)$ we mean any linear complex

[TABLE]

where each $\Delta^{i}_{\lambda}$ is a direct sum of standard modules, and $\Delta^{0}_{\lambda}\to L(\lambda)$ is an epimorphism. If such a complex happens to be exact, we say it is a BGG resolution of $L(\lambda)$.

Proposition 32**.**

Let $\Delta_{\lambda}^{\bullet}$ be a BGG complex of $L(\lambda)$.

1. $($a$)$

There exists a unique (up to scalar) non-trivial homomorphism $\xi:{\mathcal{P}}_{\lambda}^{\bullet}\to\Delta_{\lambda}^{\bullet}$ in the category of complexes. 2. $($b$)$

There exists a unique (up to scalar) non-trivial homomorphism $\eta:{\mathcal{P}}_{\lambda}^{\bullet}\to\mathcal{T}_{\lambda}^{\bullet}$ in the homotopy category. This homomorphism is a quasi-isomorphism. 3. $($c$)$

For each homomorphism $\tilde{\eta}:{\mathcal{P}}_{\lambda}^{\bullet}\to\mathcal{T}_{\lambda}^{\bullet}$ in the category of complexes descending to $\eta$ in the homotopy category, there exist a homomorphism $\varphi:\Delta_{\lambda}^{\bullet}{\longrightarrow}\mathcal{T}_{\lambda}^{\bullet}$ of complexes such that the diagram

[TABLE]

commutes in the category of complexes. If, moreover, $\Delta_{\lambda}^{\bullet}$ is exact, then such $\varphi$ is unique and injective, and $\xi$ is surjective.

Proof.

Existence parts of claims (a) and (b) follow from the fact that ${\mathcal{P}}_{\lambda}^{\bullet}$ consists of projective modules. The uniqueness part in (a) follows from the linearity of involved complexes. Indeed, since the modules in both complexes are centered at the position of their heads, from the positivity of the grading it follows that there are no non-trivial homotopies between ${\mathcal{P}}_{\lambda}^{\bullet}$ and $\Delta_{\lambda}^{\bullet}$. The uniqueness part in (b) follows from Schur’s Lemma.

To prove claim (c), note that there exists a map $\Delta^{0}_{\lambda}\to L(\lambda)$, which gives a morphism in ${\operatorname{Hom}}_{\mathcal{D}^{b}(A)}(\Delta_{\lambda}^{\bullet},\mathcal{T}_{\lambda}^{\bullet})$. Since tilting modules have costandard filtrations and since standard objects are left orthogonal to costandard objects, the existence of $\varphi$ follows from [Ha88, Lemma III.2.1].

If $\Delta_{\lambda}^{\bullet}$ is exact, then all maps in (9) are isomorphisms in $\mathcal{D}^{b}(A)$, in particular, for each $\mu\in\Lambda$, $i\in\mathbb{Z}_{\geq 0}$ and $j\in\mathbb{Z}$, the map

[TABLE]

is an isomorphism.

Let $\Delta(\mu)\langle-i\rangle$ be a direct summand of $\Delta_{\lambda}^{i}$. Then the unique up to scalar non-zero map $f:\Delta(\mu)\langle-i\rangle\to\nabla(\mu)\langle-i\rangle$ gives rise to a non-zero element in

[TABLE]

This is due to the combination of [Ha88, Lemma III.2.1], which allows us to compute extensions directly in the homotopy category of complexes and the fact that the linearity of $\Delta_{\lambda}^{\bullet}$ does not allow $f$ to be killed by any homotopy. Note also that the extension (11) can also be computed in the homotopy category using $\mathcal{T}_{\lambda}^{\bullet}$, again thanks to [Ha88, Lemma III.2.1]. And again, the positivity of the grading does not allow for any homotopies from $\mathcal{T}_{\lambda}^{\bullet}$ to $\nabla(\mu)\langle-i\rangle$. Consequently, by (10), the restriction of $\varphi$ to the summand $\Delta(\mu)\langle-i\rangle$ is uniquely defined. From this it follows that $\varphi$ is unique and injective. Similarly one sees that $\xi$ is surjective. ∎

8.3. Uniqueness of BGG resolution

Proposition 32 has the following consequence.

Theorem 33**.**

Assume that $\Delta_{\lambda}^{\bullet}$ is exact. Then all BGG resolutions of $L(\lambda)$ are isomorphic in the category of complexes.

Proof.

By Proposition 32, any BGG resolution of $L(\lambda)$ is a subcomplex of $\mathcal{T}_{\lambda}^{\bullet}$, namely, the image of the injective morphism $\varphi$ constructed in the proof. Since the Ringel dual of a balanced quasi-hereditary algebra is positively graded, see [Ma09b, Theorem 1(ii)], and both $\Delta_{\lambda}^{\bullet}$ and $\mathcal{T}_{\lambda}^{\bullet}$ are linear, there are no homotopies from $\Delta_{\lambda}^{\bullet}$ to $\mathcal{T}_{\lambda}^{\bullet}$. Therefore the morphism $\varphi$ in the proof of Proposition 32 does not depend on the choice of $\tilde{\eta}$. In particular, the image of the injective morphism $\varphi$ is a canonical subcomplex of $\mathcal{T}_{\lambda}^{\bullet}$ isomorphic to $\Delta_{\lambda}^{\bullet}$ and independent of the choice of $\Delta_{\lambda}^{\bullet}$. ∎

9. BGG complexes for ${\mathcal{S}}$-subcategories in ${\mathcal{O}}$

9.1. ${\mathcal{S}}$-subcategories in ${\mathcal{O}}$

In this section we will apply our results on singular BGG resolutions to construct analogues of BGG resolutions for certain generalization of category $\mathcal{O}$ which are no longer described by quasi-hereditary algebras but rather by the so-called standardly stratified algebras, see [FKM02], or, even, properly stratified algebras, see [Dl00]. These are the so-called ${\mathcal{S}}$-subcategories in ${\mathcal{O}}$ as defined in [FKM00]. The interest in ${\mathcal{S}}$-subcategories in ${\mathcal{O}}$ is motivated by the study of the structure of parabolically induced modules for Lie algebra, see [MS08] and references therein. As it turns out, in many cases, parabolically induced modules naturally belong to certain categories equivalent to ${\mathcal{S}}$-subcategories in ${\mathcal{O}}$. This, in particular, implies that composition multiplicities of parabolically induced modules can be described using KL-polynomials. Let us briefly recall the definition of ${\mathcal{S}}$-subcategories in ${\mathcal{O}}$.

Let $\lambda$ be a dominant integral weight and $W_{\lambda}$ the stabilizer of $\lambda+\rho$. Denote by ${}^{\lambda}\tilde{W}$ the set of longest coset representatives in $W_{\lambda}\backslash W$. Let $\mathcal{I}_{\lambda}$ denote the Serre subcategory of $\mathcal{O}_{0}$ generated by all $L(w\cdot 0)$, where $w\notin{}^{\lambda}\tilde{W}$. Then the Serre quotient category $\mathcal{S}_{\lambda}:=\mathcal{O}_{0}/\mathcal{I}_{\lambda}$ is an abelian category and we denote by $\pi:\mathcal{O}_{0}\to\mathcal{S}_{\lambda}$ the canonical projection.

Let $A$ be a basic finite dimensional associative algebra such that $A$-mod is equivalent to $\mathcal{O}_{0}$. Let $\displaystyle 1=\sum_{w\in W}e_{w}$ be a primitive decomposition of the identity in $A$, where $e_{w}$ corresponds to $L(w\cdot 0)$. Then $\mathcal{S}_{\lambda}$ is equivalent to $eAe$-mod, where

[TABLE]

The algebra $eAe$ is properly stratified in the sense of [Dl00] with $\pi(\Delta(w\cdot 0))$, where $w\in{}^{\lambda}\tilde{W}$, being a complete and irredundant list of representatives of proper standard objects with respect to the properly stratified structure. Note that, for $w_{1},w_{2}\in W$, we have $\pi(\Delta(w_{1}\cdot 0))\cong\pi(\Delta(w_{2}\cdot 0))$ if and only if $w_{1}=xw_{2}$, for some $x\in W_{\lambda}$.

From the above, it follows that the set $\{\pi(L(w\cdot 0))\,:\,w\in{}^{\lambda}\tilde{W}\}$ is a complete and irredundant list of representatives of isomorphism classes of simple objects in $\mathcal{S}_{\lambda}$.

9.2. Soergel’s equivalence

Denote by $\underline{\mathcal{O}}$ the full subcategory in $\mathcal{O}$ consisting of all modules with the property that the center of the universal enveloping algebra acts by scalars on each indecomposable direct summand. Note that all simple and all Verma modules are in $\underline{\mathcal{O}}$. For simplicity, let us restrict to the principal block of $\mathcal{O}$ and $\underline{\mathcal{O}}$.

Recall, see [Ja79, Kapitel 6], that $\mathcal{O}_{0}$ is equivalent to the category ${}^{\infty}_{\,\,\,0}\mathcal{H}_{0}^{1}$ of finitely generated Harish-Chandra $\mathfrak{g}$-$\mathfrak{g}$-bimodules having generalized central character of $\mathcal{O}_{0}$ on the left and genuine central character of $\mathcal{O}_{0}$ on the right. Under this equivalence, the subcategory $\underline{\mathcal{O}}_{0}$ corresponds to the full subcategory ${}^{1}_{0}\mathcal{H}_{0}^{1}$ of ${}^{\infty}_{\,\,\,0}\mathcal{H}_{0}^{1}$ consisting of all objects having a genuine central character on the left. As pointed out in [So86], swapping the sides of bimodules induces a self-equivalence of ${}^{1}_{0}\mathcal{H}_{0}^{1}$ and we obtain the following claim:

Proposition 34** ([So86]).**

There is an involutive self-equivalence $\Psi$ of $\underline{\mathcal{O}}_{0}$ such that $\Psi(L(w\cdot 0))\cong L(w^{-1}\cdot 0)$.

Note that $\Psi$ does not extend to ${}^{\infty}_{\,\,\,0}\mathcal{H}_{0}^{1}$ or ${\mathcal{O}}_{0}$ due to asymmetry of the requirements for the left and right actions of the center in the definition of ${}^{\infty}_{\,\,\,0}\mathcal{H}_{0}^{1}$.

9.3. BGG complexes and resolutions

A BGG complex in $\mathcal{S}_{\lambda}$ is a complex in which each component is isomorphic to a direct sum of proper standard objects. If a BGG complex has a unique homology and that homology is isomorphic to $\pi(L(w\cdot 0))$, where $w\in{}^{\lambda}\tilde{W}$, we will call such a complex a BGG resolution of $\pi(L(w\cdot 0))$. Now we can formulate our main result in this section.

Theorem 35**.**

For $w\in{}^{\lambda}\tilde{W}$, the following assertions are equivalent:

1. $($a$)$

$\pi(L(w\cdot 0))$* has a BGG resolution.* 2. $($b$)$

$L(w^{-1}\cdot\lambda)$* is a Kostant module in the sense of Subsection 6.4.*

Proof.

Assume claim (a). Let

[TABLE]

be a BGG resolution of $\pi(L(w\cdot 0))$ in $\mathcal{S}_{\lambda}$. Since all tops and all socles for all components in $Q^{\bullet}$, considered as objects in $\mathcal{O}$, are simple modules of the form $L(x\cdot 0)$, with $x\in{}^{\lambda}\tilde{W}$, homomorphisms between these components in $\mathcal{O}$ and in $\mathcal{S}_{\lambda}$ coincide. Therefore we may consider $Q^{\bullet}$ as a complex in $\mathcal{O}$. Note, however, that the fact that $Q^{\bullet}$ is exact in $\mathcal{S}_{\lambda}$ only means that, as a complex in $\mathcal{O}$, each simple subquotient of any homology in $Q^{\bullet}$ is isomorphic to $L(x\cdot 0)$, where $x\notin{}^{\lambda}\tilde{W}$.

Next we observe that all components of $Q^{\bullet}$ are, in fact, objects in $\underline{\mathcal{O}}$. Therefore we may apply the equivalence $\Psi$ to obtain a complex $\Psi(Q^{\bullet})$ of the form

[TABLE]

Each simple subquotient of any homology in $\Psi(Q^{\bullet})$ is isomorphic to $L(x\cdot 0)$, where $x\notin\tilde{W}^{\lambda}$. The latter means that translation of $\Psi(Q^{\bullet})$ to the $\lambda$-wall gives an exact complex, that is, the module $L(w^{-1}\cdot\lambda)$ has a BGG resolution. Therefore claim (b) follows from Proposition 30.

To prove that claim (b) implies claim (a), we simply reverse the above arguments. If $L(w^{-1}\cdot\lambda)$ is a Kostant module in the sense of Subsection 6.4, then $L(w^{-1}\cdot\lambda)$ has a BGG resolution by Proposition 30. Call this resolution $\hat{R}^{\bullet}$. Consider the BGG complex $R^{\bullet}$ for $L(w^{-1}\cdot 0)$. Let $\tilde{R}^{\bullet}$ be the part of $R^{\bullet}$ indexed by elements in $X_{w^{-1}}$. From Section 4, it follows that any arrow starting at an element in $X_{w^{-1}}$ goes to an element in $X_{w^{-1}}$. This implies that $\tilde{R}^{\bullet}$ is a subcomplex of $R^{\bullet}$ and we also have that $\hat{R}^{\bullet}$ is a translation of $\tilde{R}^{\bullet}$ to the $\lambda$-wall. Since $\hat{R}^{\bullet}$ is exact, each simple subquotient of any homology in $\tilde{R}^{\bullet}$ is isomorphic to $L(x\cdot 0)$, where $x\notin\tilde{W}^{\lambda}$. Now, applying $\Psi$ and then $\pi$, we get that $\pi(\Psi(\tilde{R}^{\bullet}))$ is a BGG resolution of $\pi(L(w\cdot 0))$. This completes the proof. ∎

Theorem 35 extends and corrects the main result of [FM98].

For $w\in{}^{\lambda}\tilde{W}$, let $\nabla(w)$ denote the maximal submodule of the indecomposable injective envelope of $\pi(L(w\cdot 0))$ in $\mathcal{S}_{\lambda}$ which has the property that all composition subquotients of $\nabla(w)$ (in $\mathcal{S}_{\lambda}$) are isomorphic to $\pi(L(x\cdot 0))$, where $x\in{}^{\lambda}\tilde{W}$ is such that $x\geq w$. The module $\nabla(w)$ is the costandard module corresponding to $\pi(L(w\cdot 0))$ with respect to the properly stratified structure of $\mathcal{S}_{\lambda}$.

Corollary 36**.**

Let $w\in{}^{\lambda}\tilde{W}$ be such that $\pi(L(w\cdot 0))$ is Kostant. Then, for all $x\in{}^{\lambda}\tilde{W}$ and $i\in\mathbb{Z}_{\geq 0}$, we have:

[TABLE]

Proof.

Since costandard modules are homologically right dual to proper standard modules, see [Dl00, Theoreme 5], we can use [Ha88, Lemma III.2.1] to compute the extension space in question in the homotopy category of complexes using the resolution provided by Theorem 35. The claim now follows from the explicit form of the complex (2). ∎

9.4. The right cell of the dominant weight

Projective-injective modules in $\mathcal{O}_{0}^{\lambda}$ are indexed by the elements in the right Kazhdan-Lusztig cell of $w_{0}^{\lambda}w_{0}$, see [Ir85]. The Koszul dual of this statement is that simple modules of minimal Gelfand-Kirillov dimension in $\mathcal{O}_{\lambda}$ are indexed by the left Kazhdan-Lusztig cell of $w_{0}^{\lambda}$. Applying $\Psi$, we obtain that simple modules of minimal Gelfand-Kirillov dimension in $\mathcal{S}_{\lambda}$ are indexed by the right Kazhdan-Lusztig cell of $w_{0}^{\lambda}$.

In many cases, see [IS88], parabolic category $\mathcal{O}$ has a simple projective module (any such module is also injective as $\mathcal{O}$ has a simple preserving duality). For example, this is always the case in type $A$. If a simple projective module in $\mathcal{O}^{\lambda}$ exists, one of the projective-injective modules in the regular block is obtained by translating this simple projective module from the wall to the regular block. The resulting indecomposable projective-injective module always has a predictable generalized Verma flag (just like $P(w_{0}\cdot 0)$ in $\mathcal{O}_{0}$). By Proposition 28, translation to $\mathcal{O}_{\lambda}$ of the Koszul dual simple of this indecomposable projective injective module has a BGG resolution. Therefore, by Theorem 35, the corresponding simple object in $\mathcal{S}_{\lambda}$ also has a BGG resolution. These arguments prove the following statement.

Corollary 37**.**

Assume that $\mathcal{O}^{\lambda}$ has a simple projective module. There is $w\in{}^{\lambda}\tilde{W}$ in the right Kazhdan-Lusztig cell of $w_{0}^{\lambda}$ such that $\pi(L(w\cdot 0))$ has a BGG resolution in $\mathcal{S}_{\lambda}$.

This corrects the main result of [FM98] which, in the special case $|W_{\lambda}|=2$, claimed that $\pi(L(w_{0}^{\lambda}\cdot 0))$ has a BGG resolution in $\mathcal{S}_{\lambda}$. The BGG complex constructed in [FM98] is, indeed, a complex, cf. Proposition 18, However, it is not always exact. As Subsection 7.2 shows, exactness fails, for example, in type $A_{3}$ with singularity $s_{2}$.