2.1. Setup

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

As already mentioned above, $\mathfrak{g}$ is a simple finite dimensional complex Lie algebra. We fix a triangular decomposition

(1)\begin{equation} \mathfrak{g}=\mathfrak{n}_-\oplus \mathfrak{h}\oplus \mathfrak{n}_+, \end{equation}

where $\mathfrak{h}$ is some fixed Cartan subalgebra. Denote by W the Weyl group of $\mathfrak{g}$, and by S the set of simple reflections in W, which corresponds to the triangular decomposition equation (1) above. As usual, we denote by w ₀ the longest element of W. Let ≤ denote the usual Bruhat order on W and $\ell$ the length function on W.

In this paper, we will mostly consider $\mathfrak{g}=\mathfrak{sl}_n$ with the standard triangular decomposition given by the upper triangular, the diagonal and the lower triangular matrices. In this case, we have $W\cong \mathbf{S}_n$, the symmetric group on $\{1,2,\dots,n\}$, with S being the set of elementary transpositions.

2.2. Category $\mathcal{O}$

Associated with the triangular decomposition equation (1), we have the BGG category $\mathcal{O}$ defined as the full subcategory of the category of finitely generated $\mathfrak{g}$-modules consisting of all modules on which the action of $\mathfrak{h}$ is diagonalizable and the action of $\mathfrak{n}_+$ is locally finite. We refer to [Reference Bernstein, Gelfand and Gelfand3, Reference Humphreys10] for details.

2.3. Principal block

Consider the principal block $\mathcal{O}_0$ of $\mathcal{O}$. This block is defined as the indecomposable direct summand of $\mathcal{O}$ which contains the trivial $\mathfrak{g}$-module. The isomorphism classes of simple objects in $\mathcal{O}_0$ are naturally indexed by the elements of W. For $w\in W$, we have the corresponding simple highest weight module $L_w:=L(w\cdot 0)$. Here, $0\in \mathfrak{h}^*$ denotes the zero weight and · denotes the dot action of W on $\mathfrak{h}^*$. For $\lambda\in\mathfrak{h}^*$, the dot action is defined as follows: $w\cdot \lambda=w(\lambda+\rho)-\rho$. Here, ρ is the half-sum of positive roots.

Furthermore, for $w\in W$, we have the Verma cover $\Delta_w$ of L_w and the indecomposable projective cover P_w of L_w. We denote by A the opposite of the endomorphism algebra of a multiplicity-free projective generator of $\mathcal{O}_0$. The algebra A is a finite dimensional and associative algebra. Moreover, the category A-mod of all finite dimensional A-modules is equivalent to the category $\mathcal{O}_0$.

2.4. Parabolic category $\mathcal{O}$

Let $\mathfrak{p}$ be a parabolic subalgebra of $\mathfrak{g}$ containing $\mathfrak{h}\oplus \mathfrak{n}_+$. The corresponding parabolic category $\mathcal{O}^\mathfrak{p}$ is defined as the full subcategory of $\mathcal{O}$ thatconsists of all objects on which the action of $\mathfrak{p}$ is locally finite, see [Reference Rocha-Caridi30]. We further have the principal block $\mathcal{O}_0^\mathfrak{p}$ of $\mathcal{O}^\mathfrak{p}$ defined as $\mathcal{O}_0\cap \mathcal{O}^\mathfrak{p}$.

Let $W_\mathfrak{p}$ be the parabolic subgroup of W corresponding to $\mathfrak{p}$. Denote by $W^{\mathfrak{p}}_{\mathrm{short}}$ the set of the shortest representatives in the cosets ${W_\mathfrak{p}}\hspace{-1mm}\setminus W$. The category $\mathcal{O}_0^\mathfrak{p}$ is the Serre subcategory of $\mathcal{O}_0$ generated by the simple objects L_w, where $w\in W^{\mathfrak{p}}_{\mathrm{short}}$. The category $\mathcal{O}_0^\mathfrak{p}$ is equivalent to $A_{\mathfrak{p}}$-mod for a certain quotient $A_{\mathfrak{p}}$ of A.

For $w\in W^{\mathfrak{p}}_{\mathrm{short}}$, we denote by $\Delta^\mathfrak{p}_w$ the corresponding parabolic Verma module and by $P_w^\mathfrak{p}$ the corresponding indecomposable projective module in $\mathcal{O}_0^\mathfrak{p}$.

2.5. Projective functors

Following [Reference Bernstein and Gelfand4], for every $w\in W$, there is a unique, up to isomorphism, indecomposable projective endofunctor θ_w of the category $\mathcal{O}_0$ such that $\theta_w\, P_e\cong P_w$. Any indecomposable projective endofunctor of $\mathcal{O}_0$ is isomorphic to θ_w, for some $w\in W$.

Consider the monoidal category ${\mathscr{P}}$ of all projective endofunctors of $\mathcal{O}_0$. By Soergel’s combinatorial description, see [Reference Soergel32], the monoidal category ${\mathscr{P}}$ is monoidally equivalent to the monoidal category of Soergel bimodules over the coinvariant algebra of the Weyl group W.

2.6. Twisting functors

For $w\in W$, we denote by $\top_w$ the corresponding twisting functor, see [Reference Andersen and Stroppel1, Reference Khomenko and Mazorchuk18]. For $x,y\in W$ such that $\ell(xy)=\ell(x)+\ell(y)$, we have $\top_x\Delta_y\cong \Delta_{xy}$. The main, for us, properties of twisting functors are that they functorially commute with projective functors, that they are acyclic on Verma modules and that the corresponding left derived functors are equivalences, see [Reference Andersen and Stroppel1]. In what follows, we denote by $\mathbb{L} \top_{w}$ the left derived functor of the functor $\top_w$.

2.7. Graded lift

By [Reference Soergel31], the algebra A is Koszul, and hence, it is equipped with the corresponding Koszul $\mathbb{Z}$-grading. We therefore have the category ${}^{\mathbb{Z}}\mathcal{O}_0$ of finite dimensional $\mathbb{Z}$-graded A-modules. We denote by $ \langle{}_-\rangle$ the functor which shifts the grading with the convention that $\langle 1\rangle$ shifts degree 0 to degree −1.

All structural modules in the category $\mathcal{O}_0$ admit graded lifts, see [Reference Stroppel34] for details. These graded lifts are unique, up to isomorphism and grading shift, for all indecomposable modules.

We fix standard graded lifts of all indecomposable structural modules and will use for these standard graded lifts the same notation as for the corresponding ungraded objects in $\mathcal{O}_0$, abusing notation. Similarly, we have the standard graded lifts of the indecomposable projective functors, see [Reference Stroppel34]. We denote by ${\mathscr{P}}^{\mathbb{Z}}$ the corresponding monoidal category of graded projective endofunctors of ${}^{\mathbb{Z}}\mathcal{O}_0$.

The algebra $A_{\mathfrak{p}}$ is a graded quotient of A, and we have the corresponding graded analogue ${}^{\mathbb{Z}}\mathcal{O}_0^\mathfrak{p}$ of the category $\mathcal{O}_0^\mathfrak{p}$.

2.8. Hecke algebra combinatorics

Consider the Hecke algebra $\mathbf{H}=\mathbf{H}(W,S)$ of W in the normalization of [Reference Soergel33]. It is an algebra over the commutative ring $\mathbb{Z}[v,v^{-1}]$. This algebra has the standard basis $\{H_w\,:\,w\in W\}$ as well as the Kazhdan–Lusztig basis $\{\underline{H}_w\,:\,w\in W\}$, defined in [Reference Kazhdan and Lusztig17].

The Grothendieck group $\mathbf{Gr}[{}^{\mathbb{Z}}\mathcal{O}_0]$ of the category ${}^{\mathbb{Z}}\mathcal{O}_0$ is isomorphic, as a $\mathbb{Z}[v,v^{-1}]$-module, to H by sending $[\Delta_w]$ to H_w. This isomorphism intertwines $\langle 1\rangle$ and multiplication with v ⁻¹ and sends $[P_w]$ to $\underline{H}_w$, see [Reference Beilinson and Bernstein2, Reference Brylinski and Kashiwara7].

The split Grothendieck ring $\mathbf{Gr}_\oplus[{\mathscr{P}}^{\mathbb{Z}}]$ of the monoidal category ${\mathscr{P}}^{\mathbb{Z}}$ is isomorphic to the algebra H by sending $[\theta_w]$ to $\underline{H}_w$. The defining action of ${\mathscr{P}}^{\mathbb{Z}}$ on the category $\mathcal{O}_0^{\mathbb{Z}}$ decategorifies to the right regular representation of H, see [Reference Soergel32].

Denote by $\leq_L$, $\leq_R$ and $\leq_J$ the left, right and two-sided Kazhdan-Lusztig preorders on W, respectively. The equivalence classes associated with these preorders are called left, right and two-sided cells. Each left and each right cell contains a distinguished involution, which is called the Duflo involution. For symmetric groups, all involutions are, in fact, Duflo involutions.

The singletons $\{e\}$ and $\{w_0\}$ are two-sided cells. Furthermore, we have the small two-sided cell $\mathcal{J}_s$ containing all simple reflections. In the case of the symmetric group S_n, the small cell contains exactly n − 1 left and n − 1 right cells and consists of all elements having exactly one unique expression. Multiplying the elements of this small two-sided cell with w ₀, we obtain the penultimate two-sided cell $\mathcal{J}_p$ (also referred to as the subregular cell in the literature).

The set $W^{\mathfrak{p}}_{\mathrm{short}}$ is a union of right cells. If $w_0^\mathfrak{p}$ is the longest elements of $W_\mathfrak{p}$, then $W^{\mathfrak{p}}_{\mathrm{short}}$ is the principal ideal with respect to $\leq_R$, generated by the element $w_0^\mathfrak{p} w_0$. Furthermore, for $w\in W^{\mathfrak{p}}_{\mathrm{short}}$, the projective module $P^\mathfrak{p}(w)$ is injective if and only if w and $w_0^\mathfrak{p} w_0$ belong to the same right cell. If $\mathfrak{p}$ is a minimal parabolic, that is, $W_\mathfrak{p}$ contains just one simple reflection s, the element $w_0^\mathfrak{p}w_0=sw_0$ belongs to $\mathcal{J}_p$.

2.9. Bigrassmannian permutations and socles of cokernels of inclusions of Verma modules

Recall that an element $w\in W$ is called bigrassmannian, provided that there is a unique simple reflection s such that sw < w and there is a unique simple reflection t (note that t might be equal to s) such that wt < w. By [Reference Ko, Mazorchuk and Mrdjen19, Theorem 1], in the case of the algebra $\mathfrak{sl}_n$, for $w\in \mathbf{S}_n$, the module $\Delta_e/(\Delta_w\langle -\ell(w)\rangle)$ has a simple socle if and only if w is bigrassmannian.

Furthermore, the map $w\mapsto \mathrm{Soc}(\Delta_e/(\Delta_w\langle -\ell(w)\rangle))$ is a bijection between the set of all bigrassmannian elements in S_n and graded subquotients of $\Delta_e$ of the form $L_w\langle i\rangle$, where $w\in \mathcal{J}_p$. If $w\in \mathbf{S}_n$ is not bigrassmannian, then the socle of $\Delta_e/(\Delta_w\langle -\ell(w)\rangle$ is given by the elements that correspond, under the above bijection, to those bigrassmannian elements in the Bruhat interval $[e,w]$ that are maximal among all bigrassmannian elements in this interval with respect to the Bruhat order.

3.1. Harish–Chandra bimodules

Recall from [Reference Jantzen12, Kapitel 6], that a $\mathfrak{g}$–$\mathfrak{g}$-bimodule M is called a Harish–Chandra bimodule provided that it is finitely generated as a bimodule and, additionally, that the adjoint action of $\mathfrak{g}$ on M is locally finite with finite multiplicities for composition factors. For example, for any $M\in \mathcal{O}_0$, the bimodule $U(\mathfrak{g})/\mathrm{Ann}_{U(\mathfrak{g})}(M)$ is a Harish–Chandra bimodule.

Another example can be given as follows: for two objects M and N in $\mathcal{O}_0$, the space $\mathcal{L}(M,N)$ of all linear maps from M to N on which the adjoint action of $\mathfrak{g}$ is locally finite forms a Harish–Chandra bimodule. In the case M = N, we have a natural inclusion $U(\mathfrak{g})/\mathrm{Ann}_{U(\mathfrak{g})}(M)\subset \mathcal{L}(M,M)$.

3.2. Classical Kostant’s problem

As we have already mentioned in $\S$ 1, for a $\mathfrak{g}$-module M, the corresponding Kostant’s problem, as formulated in [Reference Joseph13], is the following:

Kostant’s Problem. Is the embedding

\begin{equation*}U(\mathfrak{g})/\mathrm{Ann}_{U(\mathfrak{g})}(M)\hookrightarrow \mathcal{L}(M,M)\end{equation*}

an isomorphism?

We will denote by $\mathbf{K}(M) \in\{\mathtt{true},\mathtt{false}\}$ the logical value of the claim ‘the embedding $U(\mathfrak{g})/\mathrm{Ann}_{U(\mathfrak{g})}(M)\hookrightarrow \mathcal{L}(M,M)$ is an isomorphism’.

3.3. Kåhrström’s conjecture

The following conjecture, formulated in [Reference Ko, Mazorchuk and Mrdjen19, Conjecture 1.2], is due to Johan Kåhrström:

3.4. Kostant’s problem and 2-representation theory

Recall that the monoidal category ${\mathscr{P}}$ acts on $\mathcal{O}_0$. Given $M\in\mathcal{O}_0$, we can consider its annihilator $\mathrm{Ann}_{{\mathscr{P}}}(M)$. By definition, this is the left monoidal ideal of ${\mathscr{P}}$ consisting of all morphisms in ${\mathscr{P}}$ whose evaluation at M is zero. The quotient ${\mathscr{P}}/\mathrm{Ann}_{{\mathscr{P}}}(M)$ is, naturally, a birepresentation of ${\mathscr{P}}$.

Alternatively, the additive closure $\mathrm{add}({\mathscr{P}}\, M)$ is also a birepresentation of ${\mathscr{P}}$. Mapping

\begin{align*} {\mathscr{P}}\ni\theta\mapsto \theta(M)\in \mathrm{add}({\mathscr{P}}\, M) \end{align*}

defines an injective morphism of birepresentation from ${\mathscr{P}}/\mathrm{Ann}_{{\mathscr{P}}}(M)$ to $\mathrm{add}({\mathscr{P}}\, M)$. The arguments from [Reference Ko, Mazorchuk and Mrdjen20, Subsection 8.3] show that this morphism is an equivalence if and only if $\mathbf{K}(M)=\mathtt{true}$.

In general, finitary birepresentations of finitary bicategories can be described in terms of (co)algebra objects in a certain abelianization of the bicategory, see [Reference Mackaay, Mazorchuk, Miemietz and Tubbenhauer23]. In the case of the bicategory ${\mathscr{P}}$, the abelianization in question is the category of Harish–Chandra bimodules. Furthermore, the fact that the birepresentation $\mathrm{add}({\mathscr{P}}\, M)$ of ${\mathscr{P}}$ is given by the algebra $\mathcal{L}(M,M)$ follows from [Reference Jantzen12, Subsection 6.8]. The principal birepresentation of ${\mathscr{P}}$, which is the natural left action of ${\mathscr{P}}$ on ${\mathscr{P}}$, corresponds to the identity 1-morphism of ${\mathscr{P}}$ by [Reference Mackaay, Mazorchuk, Miemietz and Tubbenhauer23, Formula (4.2)]. In the world of Harish–Chandra bimodules, the algebra $U(\mathfrak{g})$ surjects onto the identity 1-morphism of ${\mathscr{P}}$. Consequently, the induced action of ${\mathscr{P}}$ on ${\mathscr{P}}/\mathrm{Ann}_{{\mathscr{P}}}(M)$ corresponds to the algebra given by the quotient of the identity 1-morphism of ${\mathscr{P}}$ (and hence also of $U(\mathfrak{g})$) by the annihilator of M. The algebra embedding $U(\mathfrak{g})/\mathrm{Ann}_{U(\mathfrak{g})}(M)\to \mathcal{L}(M,M)$ is a manifestation of the fact that ${\mathscr{P}}/\mathrm{Ann}_{{\mathscr{P}}}(M)$ is a sub-birepresentation of $\mathrm{add}({\mathscr{P}}\, M)$, and therefore, the positive answer to Kostant’s problem for M reformulates into the property of this natural embedding being an equivalence.

This reformulation allows us to connect the answers to Kostant’s problem for modules related by twisting functors.

We refer the reader to [Reference Mackaay, Mazorchuk, Miemietz, Tubbenhauer and Zhang24, Reference Mazorchuk and Miemietz27, Reference Mazorchuk and Miemietz28] for more details on representations of finitary bicategories.

3.5. Known results on Kostant’s problem

Below, we present a list of the results on Kostant’s problem which we could find in the existing literature.

• • In the case of symmetric groups, the value $\mathbf{K}(L_w)$ is constant on Kazhdan–Lusztig left cells, as shown in [Reference Mazorchuk and Stroppel29, Theorem 61].

• • Let $w_0^\mathfrak{p}$ denote the longest element in $W_{\mathfrak{p}}$. Then, $\mathbf{K}(L_{w_0^\mathfrak{p}w_0})= \mathtt{true}$, see [Reference Gabber and Joseph9, Theorem 4.4] and [Reference Jantzen12, Section 7.32].

• • If $s\in W^{\mathfrak{p}}$ is a simple reflection, then $\mathbf{K}(L_{sw_0^\mathfrak{p}w_0})= \mathtt{true}$, see [Reference Mazorchuk25, Theorem 1].

• • [Reference Kåhrström15, Theorem 1.1] relates the answer to Kostant’s problem for certain highest weight $\mathfrak{g}$-modules to the answer to Kostant’s problem for certain highest weight modules over the Levi quotients of $\mathfrak{p}$.

• • [Reference Kåhrström and Mazorchuk16, Theorem 1] proposes a module-theoretic characterization of the equality $\mathbf{K}(L_w)=\mathtt{true}$.

• • In [Reference Kåhrström and Mazorchuk16, Section 4], one can find a complete answer to Kostant’s problem for simple highest weight modules over $\mathfrak{sl}_n$, where $n=2,3,4,5$, and a partial answer for the same problem for $\mathfrak{sl}_6$. Some further $\mathfrak{sl}_6$-answers were computed in [Reference Kåhrström15, Section 6]. The highest weight $\mathfrak{sl}_6$-story was completed in [Reference Ko, Mazorchuk and Mrdjen20, Subsection 10.1].

• • The paper [Reference Mackaay, Mazorchuk and Miemietz22] provides a complete answer to Kostant’s problem for simple highest weight $\mathfrak{sl}_n$-modules indexed by fully commutative permutations.

4.1. Preliminaries

For $n\geq 1$, we consider the symmetric group S_n. We view elements of S_n as functions on $\underline{n}:=\{1,2,\dots,n\}$ which we compose from right to left.

Fix $k\in\{1,2,\dots,n-1\}$. For $1\leq i \lt j\leq n$, denote by $\tau^{n,k}_{i,j}$ the unique element of S_n such that

• • $\tau^{n,k}_{i,j}(i)=k$;

• • $\tau^{n,k}_{i,j}(j)=k+1$;

• • for all s < t in $\underline{n}\setminus\{i,j\}$, we have $\tau^{n,k}_{i,j}(s) \lt \tau^{n,k}_{i,j}(t)$.

We denote by $\mathcal{X}_{n,k}$ the set of all these $\tau^{n,k}_{i,j}$. Clearly, $|\mathcal{X}_{n,k}|=\binom{n}{2}$. We further split $\mathcal{X}_{n,k}$ into two disjoint subsets: $\mathcal{X}_{n,k}=\mathcal{X}_{n,k}^+\coprod\mathcal{X}_{n,k}^-$, where

\begin{align*} \mathcal{X}_{n,k}^+=\{\tau^{n,k}_{i,i+1}\,:\,i=1,2,\dots,n-1\} \quad\text{and}\quad \mathcal{X}_{n,k}^-=\mathcal{X}_{n,k}\setminus \mathcal{X}_{n,k}^+. \end{align*}

Here is an example for n = 4 and k = 2, where in the first row, we list all elements in $\mathcal{X}_{4,2}^+$, and in the second row, we list all elements in $\mathcal{X}_{4,2}^-$ (for convenience, we colour in magenta the important edges in these graphs which go to k = 2 and $k+1=3$):

We denote by G_k the centralizer of $(k,k+1)$ in S_n. The group G_k consists of all elements of S_n, which leave $\{k,k+1\}$ invariant; in particular, G_k is the direct product of the symmetric groups on $\{k,k+1\}$ and $\underline{n}\setminus\{k,k+1\}$. Hence $|G_k|=2\cdot(n-2)!$. Let $\hat{G}_k$ denote the subgroup of G_k consisting of all elements which fix both k and k + 1. Then, $\hat{G}_k$ is naturally isomorphic to the symmetric group on $\underline{n}\setminus\{k,k+1\}$ and $|\hat{G}_k|=(n-2)!$.

4.2. Formulation

For $k\in\{1,2,\dots,n-1\}$, consider the parabolic subalgebra $\mathfrak{p}=\mathfrak{p}_k$ of $\mathfrak{sl}_n$ corresponding to the simple reflection $(k,k+1)$. The composition $\circ$ in S_n gives rise to a bijection

\begin{align*} \hat{G}_k\times \mathcal{X}_{n,k} \to (\mathbf{S}_n)^{\mathfrak{p}}_{\mathrm{short}}. \end{align*}

We would like to point out the subtle difference between Theorem 3(c) and Conjecture 1(3), namely, the appearance of all possible graded shifts in Theorem 3(c). Theorem 3 provides a complete answer to Kostant’s problem for parabolic Verma modules in the above setup.

4.3. Positive cases

Let $w\in \hat{G}_k\circ\mathcal{X}_{n,k}^+$. For this w, we will prove that all assertions in Theorem 3 hold. Note that Theorem 3(e) is obvious. Also note the obvious implications Theorem 3(d) $\Rightarrow$ Theorem 3(c) $\Rightarrow$Theorem 3(b).

Assume that $w=\sigma\circ \tau^{n,k}_{i,i+1}$, for some $\sigma\in \hat{G}_k$ and some $i\in\{1,2,\dots,n-1\}$. We have the short exact sequence

\begin{align*} 0\to\Delta_{(i,i+1)}\to\Delta_e\to\Delta^{\mathfrak{p}_i}_e\to 0. \end{align*}

Since $\Delta^{\mathfrak{p}_i}_e$ is a quotient of $\Delta_e$, we have $\mathbf{K}(\Delta^{\mathfrak{p}_i}_e)=\mathtt{true}$ by [Reference Jantzen12, Section 7.32].

Note that $\tau^{n,k}_{i,i+1}\circ (i,i+1)=(k,k+1)\circ\tau^{n,k}_{i,i+1}$ and that $\sigma\circ (i,i+1)=(i,i+1)\circ \sigma$ since $\sigma\in \hat{G}_k$. From the definition of $\hat{G}_k$, it also follows that $\ell((k,k+1)w)=\ell(w)+1$. Therefore, applying $\top_{w}$ to the above short exact sequence, we obtain the isomorphisms $\top_w \Delta_{(i,i+1)}\cong\Delta_{(k,k+1)w}$ and $\top_w \Delta_e\cong \Delta_w$ and, additionally, that the non-zero map $\Delta_{(i,i+1)}\hookrightarrow\Delta_e$ is mapped to a non-zero map $\Delta_{(k,k+1)w}\to\Delta_w$, which is automatically injective as any non-zero homomorphism between the Verma modules is injective. Therefore, from the right exactness of $\top_w$, we obtain a short exact sequence

\begin{align*} 0\to\Delta_{(k,k+1)w}\to\Delta_w\to\Delta^{\mathfrak{p}}_w\to 0. \end{align*}

Since $\top_{w}$ is acyclic on Verma modules, it follows that $\Delta^{\mathfrak{p}}_w\cong \top_w \Delta^{\mathfrak{p}_i}_e \cong \mathbb{L}\top_w \Delta^{\mathfrak{p}_i}_e$. Therefore, we can now apply Lemma 2 and conclude that $\mathbf{K}(\Delta^{\mathfrak{p}}_w)=\mathtt{true}$, giving Theorem 3(a).

The non-zero objects of the form $\theta_x\, \Delta^{\mathfrak{p}_i}_e$ are exactly the indecomposable projective objects in $\mathcal{O}_0^{\mathfrak{p}_i}$. Since the latter category is a highest weight category, it has a finite global dimension, and thus, the images of the indecomposable projective objects form a basis in the Grothendieck group of this category. In particular, these images are linearly independent and therefore different.

As $\mathbb{L}\top_w$ is a derived equivalence, the classes of the non-zero modules of the form $\top_w \theta_x\, \Delta^{\mathfrak{p}_i}_e$ are also linearly independent in the corresponding Grothendieck group. In particular, they are different, which implies Theorem 3(d) (and thus also both Theorem 3(c) and Theorem 3(b), see above).

It remains to prove Theorem 3(f). The construction of twisting functors via localization (see [Reference Khomenko and Mazorchuk18]) immediately implies that $\mathrm{Ann}_{U(\mathfrak{g})}(\top_w M)\supset \mathrm{Ann}_{U(\mathfrak{g})}(M)$. The right adjoints of twisting functors, which can be realized via Enright completion functors when acting on Verma modules, have the same property. This means that $\mathrm{Ann}_{U(\mathfrak{g})}(\Delta^{\mathfrak{p}}_w)= \mathrm{Ann}_{U(\mathfrak{g})}(\Delta^{\mathfrak{p}_i}_e)$. The fact that the latter annihilator coincides with the annihilator of the (simple) socle of $\Delta^{\mathfrak{p}_i}_e$ is a standard fact, for example, it follows from [Reference Kåhrström15, Proposition 5.1]. Therefore, $\mathrm{Ann}_{U(\mathfrak{g})}(\Delta^{\mathfrak{p}_i}_e)$ (and thus also $\mathrm{Ann}_{U(\mathfrak{g})}(\Delta^{\mathfrak{p}}_w)$) is a primitive ideal.

This completes the proof of all positive cases.

4.4. Negative cases

Let $w\in \hat{G}_k\circ\mathcal{X}_{n,k}^-$. For this w, we will prove that all assertions in Theorem 3 fail. Note that $\neg$Theorem 3(e) is obvious. Also note the obvious implication $\neg$Theorem 3(b) $\Rightarrow$ $\neg$Theorem 3(d).

The argument with twisting functors used in the previous subsection reduces the present consideration to the case $w\in \mathcal{X}_{n,k}^-$, that is, $w=\tau^{n,k}_{i,j}$, for some $j\geq i+2$. We note that this w is not bigrassmannian. Let us use the results of [Reference Ko, Mazorchuk and Mrdjen19] to analyse the subquotients of $\Delta^\mathfrak{p}_{\tau^{n,k}_{i,j}}= \Delta_{\tau^{n,k}_{i,j}}/\Delta_{(k,k+1)\tau^{n,k}_{i,j}}$ of the form L_x, where $x\in \mathcal{J}_p$ (recall that $\mathcal{J}_p$ denotes the penultimate cell, see $\S$ 2.8). We will loosely call such subquotients penultimate.

To understand the penultimate subquotients in $\Delta^\mathfrak{p}_{\tau^{n,k}_{i,j}}$, we just need to compare such subquotients for the modules $\Delta_e/\Delta_{\tau^{n,k}_{i,j}}$ and for $\Delta_e/\Delta_{(k,k+1)\tau^{n,k}_{i,j}}$. By [Reference Ko, Mazorchuk and Mrdjen19, Theorem 1], the difference between these two sets is in bijection with the bigrassmannian elements in the Bruhat complement $[e,(k,k+1)\tau^{n,k}_{i,j}]\setminus [e,\tau^{n,k}_{i,j}]$. Taking into account the very explicit forms of $\tau^{n,k}_{i,j}$ and $(k,k+1)\tau^{n,k}_{i,j}$, we can determine all bigrassmannian elements in this Bruhat complement. This, however, will require consideration of a number of different cases.

To simplify our notation, we denote $\tau^{n,k}_{i,j}$ simply by τ. We also denote the elementary transposition $(l,l+1)$ by s_l.

4.5. Case 1: k = i

In this case, $\tau=s_{k+1}s_{k+2}\dots s_{j-1}$ and the Bruhat complement $[e,s_k\tau]\setminus[e,\tau]$ contains the following bigrassmannian elements: s_k, $s_{k}s_{k+1}$,…, $s_k\tau$. Note that there are at least two such elements and that they all have different right descents.

Consequently, the module $\Delta^\mathfrak{p}_{\tau^{n,k}_{i,j}}$ contains at least two (pairwise) non-isomorphic penultimate subquotients. From [Reference Ko, Mazorchuk and Mrdjen19, Proposition 22], it follows that they appear in different degrees. Let L_x and L_y be two of them (living in some degrees). Note that both x and y belong to the same Kazhdan–Lusztig right cell, namely, the unique right cell inside $\mathcal{J}_p$, which indexes some simples in $\mathcal{O}_0^{\mathfrak{p}}$. Since x ≠ y, these two elements belong to different Kazhdan–Lusztig left cells. In particular, from [Reference Mazorchuk and Miemietz27, Lemma 12], it follows that $\theta_{x^{-1}}$ kills all simples in $\mathcal{O}_0^{\mathfrak{p}}$ except for L_x, while $\theta_{y^{-1}}$ kills all simples in $\mathcal{O}_0^{\mathfrak{p}}$ except for L_y.

Therefore, up to graded shift, $\theta_{x^{-1}}\Delta^{\mathfrak{p}}_w$ is isomorphic to $\theta_{x^{-1}}L_x$. Similarly, $\theta_{y^{-1}}\Delta^{\mathfrak{p}}_w$ is isomorphic to $\theta_{y^{-1}}L_y$. However, we have $\theta_{x^{-1}}L_x\cong P^\mathfrak{p}_d\cong \theta_{y^{-1}}L_y$, where d is the Duflo involution in the right cell of x, see [Reference Mazorchuk26, Section 3]. Note that the latter cell coincides with the right cell of y. This shows that Theorem 3(b), Theorem 3(c) and Theorem 3(d) fail.

Since Theorem 3(b) fails, the birepresentations ${\mathscr{P}}/\mathrm{Ann}_{{\mathscr{P}}}(\Delta^{\mathfrak{p}}_w)$ and $\mathrm{add}({\mathscr{P}}\, \Delta^{\mathfrak{p}}_w)$ cannot be equivalent, and hence, Theorem 3(a) fails as well.

Finally, since L_x and L_y belong to different left cells inside the same two-sided cell, their annihilators are incomparable primitive ideals of minimal Gelfand–Kirillov dimension (among all other primitive ideals for $\mathcal{O}_0^\mathfrak{p}$). Therefore, the annihilator of $\Delta_w^\mathfrak{p}$ must be contained in the intersection of these two ideals and hence cannot be a primitive ideal. This shows that Theorem 3(f) fails and completes Case 1.

4.6. Case 2: $k+1=j$

This case follows from Case 1 using the symmetry of the root system.

4.7. Case 3: i > k

In this case, $\tau=s_{k+1}s_{k+2}\dots s_{j-1}s_{k}s_{k+1}\dots s_{i-1}$ and the Bruhat complement $[e,s_k\tau]\setminus[e,\tau]$ contains the following bigrassmannian elements: $s_{k}s_{k+1}\dots s_{i}$, $s_ks_{k+1}\dots s_{i+1}$…, $s_ks_{k+1}\dots s_{j-1}$. Note that there are at least two such elements (since $j\neq i+1$). Applying to them the arguments from Case 1, we complete the proof.

4.8. Case 4: $j \lt k+1$

This case follows from Case 3 using the symmetry of the root system.

4.9. Case 5: i < k and $j \gt k+1$

In this case, $\tau=s_{k-1}s_{k-2}\dots s_{i}s_{k+1}s_{k+2}\dots s_{j-1}$ and the Bruhat complement $[e,s_k\tau]\setminus[e,\tau]$ contains the following bigrassmannian elements: s_k, $s_{k}s_{k+1}$,…, $s_ks_{k+1}\dots s_{j-1}$, $s_ks_{k-1}$,… $s_ks_{k-1}\dots s_{i}$. Clearly, there are at least two such elements. Applying to them the arguments from Case 1, we complete the proof.

4.10. Asymptotic

From Theorem 3, we see that the ratio of positive vs negative cases equals

\begin{align*} \frac{|\mathcal{X}_{n,k}^+|}{|\mathcal{X}_{n,k}^-|} = \frac{n-1}{\binom{n}{2}-(n-1)}. \end{align*}

This goes to 0 when n goes to infinity, which aligns with the results of [Reference Mackaay, Mazorchuk and Miemietz22, Section 6]. It is also an interesting observation that this ratio does not depend on k.

5.1. Setup

For $1\leq k \lt n$, set $m=n-k$. Let $\mathfrak{q}=\mathfrak{q}_k$ be the unique parabolic subalgebra of $\mathfrak{sl}_n$ whose Levi factor is $\mathfrak{sl}_k\oplus\mathfrak{sl}_m$, with $\mathfrak{sl}_k$ adjusted at the top left corner. This means that the only simple reflection that is missed by $\mathfrak{q}_k$ is $(k,k+1)$.

5.2. Weights and oriented cup diagrams

We will now present a concise version of the combinatorial diagrammatic description of the category $\mathcal{O}_0^\mathfrak{q}$ from [Reference Brundan and Stroppel5, Reference Brundan and Stroppel6], so we refer the reader to these paper for all technical details.

We denote by $\mathcal{W}_{n,k}$ the set of all words of length n in the alphabet with two letters, $\vee$ and $\wedge$, in which the letter $\wedge$ appears exactly k times. The letter $\vee$ should be thought of as the head of an arrow pointing down, while the letter $\wedge$ should be thought of as the head of an arrow pointing up. For example, we have

\begin{align*} \mathcal{W}_{4,2}=\{\wedge\wedge\vee\vee,\,\, \wedge\vee\wedge\vee,\,\,\wedge\vee\vee\wedge,\,\, \vee\wedge\wedge\vee,\,\,\vee\wedge\vee\wedge,\,\,\vee\vee\wedge\wedge \}. \end{align*}

We call the unique word in $\mathcal{W}_{n,k}$ which starts with k wedges (i.e., $\wedge$) dominant. Hence, in the above example, the dominant word is $\wedge\wedge\vee\vee$.

The group S_n acts on $\mathcal{W}_{n,k}$ by permuting the positions of the letters in a word. This action induces a natural bijection Φ between $W^\mathfrak{q}_\mathrm{short}$ and $\mathcal{W}_{n,k}$, which sends $w\in W^\mathfrak{q}_\mathrm{short}$ to the image of the dominant word under w ⁻¹.

Given an element $\lambda\in\mathcal{W}_{n,k}$, we can form an oriented cup diagram by attaching to letters of λ ends of vertically falling strings and cups that altogether form a planar and oriented diagram. In particular, the requirement to be oriented means that the two ends of a cup should be attached to different letters (i.e., one to a $\wedge$ and the other one to a $\vee$). For example, here are the six oriented cup diagrams that can be drawn for the elements $\wedge\vee\wedge\vee$:

The degree of an oriented cup diagram is the number of cups oriented clockwise.

5.3. Standard modules

An oriented cup diagram is called admissible, provided that this diagram does not contain any vertical strand under $\vee$ to the left of any vertical strand under $\wedge$. For example, for the element $\wedge\vee\wedge\vee$, out of the six oriented cup diagrams above, all but the first one (i.e., but the one with four vertical strands) are admissible.

For each $\lambda\in\mathcal{W}_{n,k}$, there is a unique admissible oriented cup diagram $\mathbf{d}(\lambda)$ for λ of degree zero. It can be constructed recursively using the following algorithm. If λ does not contain any $\vee$ to the left of some $\wedge$, then $\mathbf{d}(\lambda)$ consists of vertical strings. Otherwise, λ must contain a subword $\vee\wedge$ which we connect by a cup which becomes oriented counterclockwise. We can now remove this cup and proceed recursively.

Conversely, given an unoriented cup diagram d with at most $\min(k,m)$ cups, there is a unique $\lambda\in\mathcal{W}_{n,k}$ such that, removing the orientation from $\mathbf{d}(\lambda)$, we obtain d. In order to construct λ, we orient each cup in d counterclockwise. After orienting cups, the admissibility condition will uniquely determine positions of the remaining $\wedge$s and $\vee$s. For example, if n = 7 and k = 3, then, for the diagram,

we get the corresponding element $\vee\vee\wedge\wedge\vee\vee\wedge\in\mathcal{W}_{7,3}$.

The following proposition is a reformulation (with adaptation to our terminology) of [Reference Brundan and Stroppel5, Theorem 5.1].

Let us consider the example of n = 4 and k = 2. In this example, we have

\begin{align*} W^\mathfrak{q}_\mathrm{short}= \{e,s_2,s_2s_1,s_2s_3,s_2s_1s_3,s_2s_1s_3s_2\}. \end{align*}

For the element x = e, here are the corresponding admissible oriented cup diagrams:

The corresponding degrees, from left to right, are 0, 1 and 2. For each underlying unoriented cup diagram, we can now write the corresponding words for which the diagram becomes of degree zero:

We see that the first word corresponds to e, the second to s ₂ and the third to $s_2s_1s_3s_2$. We conclude that $\Delta_e^\mathfrak{q}$ has L_e in degree 0, and then it has $L_{s_2}$ in degree 1 and, finally, $L_{s_2s_1s_3s_2}$ in degree 2.

For the element $x=s_2$, we saw five admissible oriented cup diagrams in $\S$ 5.2. From this, we conclude that $\Delta^\mathfrak{q}_{s_2}$ has $L_{s_2}$ in degree 0, then $L_{s_2s_1}$, $L_{s_2s_3}$ and $L_{s_2s_1s_3s_2}$ in degree 1 and, finally, $L_{s_2s_1s_3}$ in degree 2.

5.4. Thin standard modules

Set $\mathtt{a}=\min(k,m)$. For $w\in W^\mathfrak{q}_\mathrm{short}$, we will say that the module $\Delta^\mathfrak{q}_w$ is thin, provided that there is a unique $y\in W^\mathfrak{q}_\mathrm{short}$ such that

• • L_y is a subquotient of $\Delta^\mathfrak{q}_w$;

• • the diagram $\mathbf{d}(\Phi(y))$ contains exactly $\mathtt{a}$ cups.

We note that, for $w\in W^\mathfrak{q}_\mathrm{short}$, the number of cups in $\mathbf{d}(\Phi(w))$ coincides with the value of Lusztig’s a-function on w, see [Reference Lusztig21]. Hence, the value $\mathtt{a}$ is the maximum value which the a-function attains at the elements of $W^\mathfrak{q}_\mathrm{short}$. By a result of Irving, see [Reference Irving11, Proposition 4.3], any socular constituent L_y of any $\Delta^\mathfrak{q}_w$, where $w\in W^\mathfrak{q}_\mathrm{short}$, has the property that the diagram $\mathbf{d}(\Phi(y))$ contains exactly $\mathtt{a}$ cups. Consequently, a thin parabolic Verma module must have simple socle.

It is easy to check, by a direct computation, that in our running example n = 4 and k = 2, the list of all thin parabolic Verma modules looks as follows:

\begin{align*} \Delta^\mathfrak{q}_e,\quad \Delta^\mathfrak{q}_{s_2s_1},\quad \Delta^\mathfrak{q}_{s_2s_3},\quad \Delta^\mathfrak{q}_{s_2s_1s_3s_2}. \end{align*}

5.5. Formulation

Let n and k be as above. Denote by $\mathcal{Y}_{n,k}$ the set of all elements $w\in W^\mathfrak{q}_\mathrm{short}$ such that the word $\Phi(w)$ has the following property:

• • if $k \lt n-k$, then all $\vee$s in w appear next to each other;

• • if $k \gt n-k$, then all $\wedge$s in w appear next to each other;

• • if $k=n-k$, then either all $\vee$s or all $\wedge$s in w appear next to each other.

For example, if n = 5 and k = 2, then $k=2 \lt 3=n-k$ and

\begin{align*} \mathcal{Y}_{5,2}=\{\wedge\wedge\vee\vee\vee,\quad\wedge\vee\vee\vee\wedge,\quad \vee\vee\vee\wedge\wedge\}. \end{align*}

At the same time, if n = 4 and k = 2, then $k=n-k$ and we have

\begin{align*} \mathcal{Y}_{4,2}=\{\wedge\wedge\vee\vee,\quad\wedge\vee\vee\wedge,\quad \vee\wedge\wedge\vee,\quad\vee\vee\wedge\wedge\}. \end{align*}

We can now formulate our main result for maximal parabolic subalgebras.

We note that the case $k\in\{1,n-1\}$ is covered by both statements of Theorem 5.

The next two subsections are dedicated to prove Theorem 5.

5.6. Non-thin parabolic Verma modules are Kostant negative

If $\Delta_w^\mathfrak{q}$ is not thin, for some $w\in W^\mathfrak{q}_\mathrm{short}$, then there exist different $x,y\in W^\mathfrak{q}_\mathrm{short}$ of maximal a-value such that both L_x and L_y are subquotients of $\Delta_w^\mathfrak{q}$ (necessarily with multiplicity 1). Let d be the unique involution in the Kazhdan–Lusztig right cell of x (which coincides with the corresponding cell for y). Then,

\begin{align*} \theta_{x^{-1}}\Delta_w^\mathfrak{q}\cong \theta_{x^{-1}}L_x\cong \theta_dL_d\cong \theta_{y^{-1}}L_y\cong \theta_{y^{-1}}\Delta_w^\mathfrak{q}. \end{align*}

Since $\theta_{x^{-1}}\not\cong\theta_{y{-1}}$ while $\theta_{x^{-1}}\Delta_w^\mathfrak{q}\cong \theta_{y^{-1}}\Delta_w^\mathfrak{q}$, from $\S$ 3.4, it follows that $\mathbf{K}(\Delta_w^\mathfrak{q})=\mathtt{false}$.

5.7. Kostant’s problem for thin parabolic Verma modules

After the observation in the previous subsection, we are left to consider thin parabolic Verma modules. We start with a classification of such modules.

Now, we need to consider two cases. Recall that, to avoid trivial cases, we assume $1\leq k\leq n-1$. Let w be such that $\lambda=\Phi(w)$. Note that if the flip number of λ is equal to 2, then either w = e or $w=w_0^{\mathfrak{q}}w_0$. In the first case, the corresponding parabolic Verma module is a quotient of the projective Verma module, and hence, it is Kostant positive. In the second case, the corresponding parabolic Verma module is simple and Kostant positive by $\S$ 3.5, second bullet. Thus, we assume that the flip number of λ is equal to 3.

Case 1. Assume $k=\frac{n}{2}$. In this case, we have $a_2=a_1+a_3$. This implies that any admissible oriented cup diagram for λ with k cups must have two non-nested cups next to each other. By the main result of [Reference Mackaay, Mazorchuk and Miemietz22], the corresponding simple module L_u, which is the socle of our parabolic Verma module, is Kostant negative.

Let us now analyse the module $\Delta^\mathfrak{q}_w$ in more detail. There is a unique admissible oriented cup diagram for λ with k cups. Any other admissible oriented cup diagrams for λ are obtained from this one by replacing some outer clockwise oriented cups by vertical strings. Here is an example, for n = 8 and k = 4, of an original cup diagram

and the cup diagrams that can be obtained from it by replacing some outer clock-wise oriented cups:

In [Reference Mackaay, Mazorchuk and Miemietz22, Section 5.5], one can find an explicit construction of two different elements x and y in W such that $\theta_x L_u\cong \theta_y L_u$. In the case of the above (generic) example, these elements x and y are given by the following diagrams:

Note the outer right lower cup in x. For all admissible oriented cup diagrams for λ, except the original one, this cup hits vertical strands. This means that θ_x kills the corresponding simple module; in other words, $\theta_x L_u\cong \theta_x\Delta_w^\mathfrak{q}$. Similar arguments apply to general w leading to the same conclusion $\theta_x L_u\cong \theta_x\Delta_w^\mathfrak{q}$.

Since $\theta_x L_u\cong \theta_y L_u$ and $\theta_y L_u$ is a submodule of $\theta_y\Delta_w^\mathfrak{q}$, we thus have a non-zero degree zero morphism from $\theta_x\Delta_w^\mathfrak{q}$ to $\theta_y\Delta_w^\mathfrak{q}$. As θ_x and θ_y are indecomposable and non-isomorphic and the endomorphism algebra of projective functors is positively graded, it follows that this degree zero zero map from $\theta_x\Delta_w^\mathfrak{q}$ to $\theta_y\Delta_w^\mathfrak{q}$ cannot be the evaluation of some map from θ_x to θ_y at $\Delta_w^\mathfrak{q}$. From $\S$ 3.4, we therefore conclude that the answer to Kostant’s problem for $\Delta_w^\mathfrak{q}$ is negative.

Case 2. Assume $k\neq \frac{n}{2}$. In this case, we have $a_2 \gt a_1+a_3$. This implies that any oriented cup diagram for λ with $\min(k,n-k)$ cups must have a vertical strand that separates the two sets of nested cups. By the main result of [Reference Mackaay, Mazorchuk and Miemietz22], the corresponding simple module L_u, which is the socle of our parabolic Verma module, is Kostant positive. For different x and y in W, we have an injective restriction map from $\mathrm{Hom}_\mathfrak{g}(\theta_x\Delta_w^\mathfrak{q},\theta_y\Delta_w^\mathfrak{q})$ to $\mathrm{Hom}_\mathfrak{g}(\theta_xL_u,\theta_yL_u)$.

Since L_u is Kostant positive, $\mathrm{Hom}_{{\mathscr{P}}}(\theta_x,\theta_y)$ surjects on the latter and hence also on the former. This implies that $\Delta_w^\mathfrak{q}$ is Kostant positive.

This completes the proof.

5.8. Example: the extreme maximal parabolic

Consider the parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{sl}_n$ corresponding to the choice of all simple roots but the first one. Then, the Levi factor of $\mathfrak{p}$ is isomorphic to $\mathfrak{sl}_{n-1}$. Setting $s_i=(i,i+1)$, we have

\begin{align*} W^{\mathfrak{p}}_{\mathrm{short}}= \{e,s_1,s_1s_2,\dots,s_1s_2s_{3}\cdots s_{n-1}\}. \end{align*}

It is well-known, see, for example, [Reference Stroppel35], that the corresponding category $\mathcal{O}_0^\mathfrak{p}$ is equivalent to the category of modules over the following quiver

with the following relations:

\begin{align*} \alpha_1\beta_1=0,\quad \alpha_i\alpha_{i+1}=0,\quad \beta_i\beta_{i-1}=0,\quad \alpha_i\beta_i=\beta_{i-1}\alpha_{i-1}. \end{align*}

Here, the vertex 0 corresponds to e, and, for i > 0, the vertex i corresponds to $s_1s_2\cdots s_i$.

In this relation, the parabolic Verma modules are the standard modules with the (unique) quasi-hereditary structure.

The a-value of e is zero, while the a-value of all other elements in $(\mathbf{S}_n)^\mathfrak{p}_\mathrm{short}$ is 1. It follows that the only thin parabolic Verma modules are Δ₀ and $\Delta_{n-1}$. From Theorem 5, we obtain that these two parabolic Verma modules are the only Kostant positive parabolic Verma modules in this case.

In fact, in this case, one can extend Theorem 5 in the following way (which is an analogue of Theorem 3).

5.9. Asymptotic

From Theorem 5, we see that the ratio of positive vs negative cases equals $\frac{2}{\binom{n}{k}-2}$, for $k=1,n-1,\frac{n}{2}$. For $k\neq \frac{n}{2}$, this ratio equals

\begin{align*} \frac{\min(k,n-k)+1}{\binom{n}{k}-(\min(k,n-k)+1)}. \end{align*}

This goes to 0 when n goes to infinity, which again aligns with the results of [Reference Mackaay, Mazorchuk and Miemietz22, Section 6].

6.1. Preliminary definitions

For a composition $\mu\models n$, we denote by $\mathbf{c}(\mu)\vdash n$ the corresponding partition of n. The partition $\mathbf{c}(\mu)$ is obtained from µ by ordering the parts of the latter in a weakly decreasing order. For two compositions $\mu,\nu\models n$, we write $\mu\sim\nu$, provided that $\mathbf{c}(\mu)=\mathbf{c}(\nu)$.

There is a natural bijection between compositions of n and parabolic subalgebras of $\mathfrak{sl}_n$ given as follows, for $\mu=(\mu_1,\mu_2,\dots)$, the corresponding subalgebra $\mathfrak{p}_\mu$ has diagonal blocks of size $\mu_1,\mu_2,\dots$ reading along the main diagonal from north-west to south-east.

For $\mu=(\mu_1,\mu_2,\dots,\mu_k)\models n$, define the sets

\begin{align*} X_1=\{1,2,\dots,\mu_1\}, X_2=\{\mu_1+1,\mu_1+2,\dots,\mu_1+\mu_2\} \quad\text{and so on}. \end{align*}

We will call these sets the blocks of µ. Denote by G_µ the subgroup of S_n consisting of all permutations π satisfying the following property: for each $i\in\{1,2,\dots,k\}$, there is some $j\in\{1,2,\dots,k\}$ such that $|X_i|=|X_j|$ and π sends the elements of X_i to the elements of X_j preserving the natural order between these elements. Here is an example of an element in $G_{(2,2,1,2)}$, where $X_1=\{1,2\}$, $X_2=\{3,4\}$, $X_3=\{5\}$ and $X_4=\{6,7\}$:

Let $\mu,\nu\models n$ be two compositions such that $\mu\sim\nu$. Let $X_1,X_2,\dots,X_K$ be the blocks of µ and $Y_1,Y_2,\dots,Y_K$ be the blocks of ν. There is a unique $\tau\in S_k$ such that

• • $|X_i|=|Y_{\tau(i)}|$, for all i;

• • if i < j and $|X_i|=|X_j|$, then $\tau(i) \lt \tau(j)$.

We denote by $\omega_{\mu,\nu}$ the unique element of S_n which sends, for each i, the elements of X_i to the elements of $Y_{\tau(i)}$ preserving the natural order among these elements. For example,

note that $\omega_{\mu,\nu}G_\mu=G_\nu\omega_{\mu,\nu}$.

The parabolic subgroup W_µ of S_n corresponding to the composition µ is the product $S_{X_1}\times S_{X_2}\times\dots\times S_{X_k}$ of symmetric groups.

6.2. Some positive cases

For $\mu\models n$, consider $\mathfrak{p}=\mathfrak{p}_\mu$.

Note that, if ν and $\nu^\prime$ are different compositions of n such that $\mu\sim\mu$ and $\nu^\prime\sim\mu$, then the sets $G_\mu\omega_{\nu,\mu}$ and $G_\mu\omega_{\nu^\prime,\mu}$ are disjoint.

6.3. Some negative cases

For $\mu\models n$, consider $\mathfrak{p}=\mathfrak{p}_\mu$. Let $w_0^\mu$ be the longest element in the parabolic subgroup of W corresponding to $\mathfrak{p}$. Let $\mathcal{R}$ denote the right Kazhdan–Lusztig cell of W containing the element $w_0^\mu w_0$.

For $w\in (\mathbf{S}_n)_\mathrm{short}^\mathfrak{p}$, we say that the parabolic Verma module $\Delta_w^\mathfrak{p}$ is thin, provided that there is a unique $u\in \mathcal{R}$ such that $[\Delta_w^\mathfrak{p}:L_u]\neq 0$; moreover, $[\Delta_w^\mathfrak{p}:L_u]=1$.

The following proposition is a general off-shot from the arguments applied in $\S$ 5.

6.4. Some speculations

The above results reduce the study of Kostant problem for the parabolic Verma module to the case of thin parabolic Verma modules that are not covered by $\S$ 6.2. Each thin parabolic Verma module has simple socle. The results of $\S$ 5 suggest that, for a thin parabolic Verma module $\Delta_w^\mathfrak{p}$ with socle L_u, we have $\mathbf{K}(\Delta_w^\mathfrak{p})=\mathbf{K}(L_u)$. We do not know whether this is true in general and if yes, how to prove this. In any case, this is only a reduction to the still open problem of determining $\mathbf{K}(L_u)$.

The above results also suggest that classification of thin parabolic Verma modules is an interesting and important problem.