KOSTANT MODULES IN BLOCKS OF CATEGORY O_{S}

BRIAN D. BOE AND MARKUS HUNZIKER

Abstract. In this paper the authors investigate infinite-dimensional representations L in blocks of the relative (parabolic) category O_{S} for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in L “looks like” the cohomology with coefficients in a finite-dimensional module, as in Kostant’s theorem. A complete classification of these “Kostant modules” in regular blocks for maximal parabolics in the simply laced types is given. A complete classification is also given in arbitrary (singular) blocks for Hermitian symmetric categories.

1. Introduction

1.1. Let 𝔤 be a complex semisimple Lie algebra with a standard parabolic subalgebra 𝔭_{S} = 𝔪_{S} + 𝔲_{S}, where 𝔲_{S} is the nilradical and 𝔪_{S} the Levi subalgebra, corresponding to a subset S of the simple roots of 𝔤. Let W be the Weyl group of 𝔤 and W_{S} the parabolic subgroup of W corresponding to S. In [Kos], Kostant proved that for any finite-dimensional simple 𝔤-module E, the Lie algebra cohomology H^{∗}(𝔲_{S}, E) is multiplicity-free as an 𝔪_{S}- module, and that the decomposition is described by the graded poset ^{S}W of minimal length coset representatives of W_{S}\W. Explicitly, if E = E_{λ} is a simple finite-dimensional 𝔤-module of highest weight λ, then as an 𝔪_{S}-module,

(1.1.1)  H^{i}(𝔲_{S}, E_{λ}) ≃ ⨁_{\begin{matrix}y∈^{S}W \\ ℓ(y)=i \end{matrix}} F_{y(λ+ρ)−ρ},

where ℓ(y) is the length of y ∈ ^{S}W and F_{y(λ+ρ)−ρ} is a simple 𝔪_{S}-module of highest weight y(λ + ρ) − ρ. Here ρ is half the sum of the positive roots of 𝔤 as usual.

1.2. In this paper we investigate infinite-dimensional simple highest weight modules of 𝔤 that have 𝔲_{S}-cohomology formulas analgous to Kostant’s formula (1.1.1). We will work in the relative (parabolic) Bernstein-Gelfand-Gelfand category O_{S} that was introduced by Rocha-Caridi [RC]. The relative category O_{S} can be decomposed into a direct sum of simpler blocks, each block being a full subcategory consisting of modules with the same central character. Each block contains only finitely many simple modules. Regular blocks, corresponding to regular central characters, are the easiest to describe. The simple modules in a regular block are in one-to-one correspondence with the poset ^{S}W. For w ∈ ^{S}W, we will write L_{w} for the corresponding simple module.

Date: April 14, 2006.

Research of the first author partially supported by NSA grant H98230-04-1-0103.

The precise definition of a Kostant module will be given in Section 3. In fact we will give two definitions. First we define what we call weak Kostant modules. These modules are the simple highest weight modules for which the associated non-zero Kazhdan-Lusztig polynomials are all one. This condition turns out to be equivalent to the 𝔲-cohomology being given by a formula like (1.1.1), but with ^{S}W replaced by a more general ranked poset. Kostant modules are then weak Kostant modules for which the 𝔲-cohomology does precisely look like the 𝔲-cohomology of a finite dimensional module.

1.3. Acknowledgements. The notion of a Kostant module is not entirely new. In [Col], Collingwood introduced the notion of Kostant module for Harish-Chandra modules and classified what in our terminology would be weak Kostant modules with regular infinitesimal character for Hermitian symmetric pairs. Also in the Hermitian symmetric pair setting, Enright in [Enr] proved a 𝔲-cohomology formula analogous to Kostant’s formula for all unitary highest weight modules (with not necessarily regular infinitesimal character) by using methods that were developed earlier by Enright and Shelton in [ES1]. The authors thank Daniel Nakano and Jonathan Kujawa for helpful discussions.

2. Notation and Preliminaries

2.1. Notation. In the following let 𝔤 be a (finite-dimensional) complex simple Lie algebra with fixed Cartan subalgebra 𝔥 and Borel subalgebra 𝔟 containing 𝔥. Let Φ ⊂ 𝔥^{∗} denote the root system of (𝔤, 𝔥). For α ∈ Φ, let 𝔤^{α} denote the root subspace of 𝔤 corresponding to α. If 𝔞 is an 𝔥-invariant subspace of 𝔤, let Φ(𝔞) = {α ∈ Φ : 𝔤^{α} ⊂ 𝔞}. Let Φ^{+} = Φ(𝔟) be the set of positive roots and let Δ be the subset of simple roots in Φ^{+}. Let ρ be half the sum of all the positive roots. Every subset S ⊂ Δ determines in the usual way a standard parabolic subalgebra 𝔭_{S} = 𝔪_{S} ⊕ 𝔲_{S} containing 𝔟 with Levi factor 𝔪_{S} and nil radical 𝔲_{S} such that Φ(𝔪_{S}) ∩ Δ = S and Φ(𝔲_{S}) ∩ Δ = Δ \ S. We will frequently specify a parabolic subalgebra 𝔭_{S} by giving the pair (Φ, Φ_{S}), where Φ_{S} = Φ(𝔪_{S}). Define the Dynkin diagram of the pair (𝔤, 𝔭_{S}) as the Dynkin diagram of 𝔤 with the nodes corresponding to the simple roots in Δ \ S crossed. By the type of such a diagram we will mean the pair of types of (Φ, Φ_{S}). For example, Figure 1 shows the diagram of type (A_{3}, A_{1} × A_{1}) of the simple Lie

a  b  c

Figure 1. Dynkin diagram for (A_{3}, A_{1} × A_{1})

algebra of type A_{3} and its standard parabolic subalgebra corresponding to S = {a, c}.

2.2. Category O_{S}. Fix S ⊂ Δ and let 𝔭 = 𝔪 ⊕ 𝔲 be the standard parabolic corresponding S. (We will usually omit the subscript S when S has been fixed.) Let O_{S} be the category of all finitely generated 𝔤-modules on which 𝔪 acts finitely semisimply and 𝔲 acts locally nilpotently. If S = ∅ (i.e., if 𝔭 = 𝔟), then the category O_{S} is the ordinary Bernstein-Gelfand-Gelfand category O. The most basic objects in O_{S} are the generalized Verma modules. Let F_{μ} be a finite-dimensional simple 𝔪-module with highest weight μ. We may view F_{λ} as a 𝔭-module by letting the nilradical 𝔲 act trivially. Then the generalized Verma module (GVM) with highest weight λ is the induced module

N_{λ} = U(𝔤) ⊗_{U(𝔭)} F_{λ}.

The module N_{λ} is a quotient of the ordinary Verma module M_{λ} with highest weight λ. Let L_{λ} denote the simple quotient of both N_{λ} and M_{λ}. The Verma module M_{λ} and all of its subquotients, in particular N_{λ} and L_{λ}, admit a central character which we denote χ_{λ}. Recall that this means that χ_{λ} : Z(U(𝔤)) → ℂ is a character of the center of U(𝔤) such that z.m = χ_{λ}(z)m for all z ∈ Z(U(𝔤)) and all m in the module. For any character χ : Z(U(𝔤)) → ℂ let O_{S}^{χ} denote the full subcategory of O_{S} consisting of modules V such that z − χ(z) acts locally nilpotently on V for all z ∈ Z. Then the category O_{S} decomposes as

O_{S} = ⨁_{χ∈Z(U(𝔤))^{∧}} O_{S}^{χ}.

Here O_{S}^{χ} is non-zero if and only if χ is of the form χ = χ_{λ} for some S-dominant S-integral λ ∈ 𝔥^{∗}. In the following, if χ = χ_{λ} we will write O_{S}^{λ} instead of O_{S}^{χ}. The categories O_{S}^{λ} are called the blocks of O_{S}. Let W denote the Weyl group of 𝔤. As usual we write w · λ for w(λ + ρ) − ρ. By a theorem of Harish-Chandra, χ_{λ} = χ_{μ}, and hence O_{S}^{λ} = O_{S}^{μ}, if and only if λ ∈ W · μ. Thus, without loss of generality, we may consider only blocks of the form O_{S}^{μ}, where μ + ρ is anti-dominant. Furthermore, for simplicity, we will only consider integral weights. So, from now on we will assume that μ + ρ is an anti-dominant integral weight, i.e., (μ + ρ, α^{∨}) is a non-positive integer for all α ∈ Δ. Here ( , ) denotes the bilinear form on 𝔥^{∗} (induced from the Killing form) and α^{∨} = 2α/(α, α).

2.3. Posets and parametrization of simple modules in O_{S}^{μ}. Let W_{S} be the subgroup of W generated by the simple reflections s_{α} with α ∈ S. Define

SW = {w ∈ W | l(s_{α}w) = l(w) + 1 for all α ∈ S},

where l is the usual length function on W. Then ^{S}W is the set of minimal length coset representatives for W_{S}\W. There is a natural ordering on ^{S}W which is induced by the Bruhat ordering on W (cf. [Hum]). We view ^{S}W as a poset (partially ordered set) with respect to this ordering and we will later often draw a Hasse diagram of ^{S}W with the least element e at the bottom (e.g., Figures 2, 3). We sometimes label an edge x—w by a simple root α if w = xs_{α} (as in Figure 3).

Now let μ ∈ 𝔥^{∗} be such that μ + ρ is anti-dominant integral and let w_{S} be the longest element of W_{S}. Then every highest weight λ of a highest weight module in O_{S}^{μ} can be written in the form λ = w_{S}w · μ for some w ∈ ^{S}W. If μ + ρ is not regular then the element w ∈ ^{S}W is not unique. Define the set of singular simple roots by

J = {α ∈ Δ | (μ + ρ, α^{∨}) = 0}.

Then W_{J} = W_{μ} = {w ∈ W | w · μ = μ}. Define

SW^{J} = {w ∈ ^{S}W | w < ws_{α} ∈ ^{S}W for all α ∈ J}.

In the Hasse diagram of ^{S}W, a node that corresponds to an element of ^{S}W^{J} is a node having an edge with label α going up from it for every α ∈ J. For now we will view ^{S}W^{J} as a poset with ordering induced from the Bruhat ordering on ^{S}W. (A different ordering will be introduced later in Section 7.)

Every highest weight λ of a highest weight module in O_{S}^{μ} can be written uniquely in the form λ = w_{S}w · μ for some w ∈ ^{S}W^{J}. In the following we will write L_{w} (resp. N_{w}) for the simple module (resp. GVM) in O_{S}^{μ} of highest weight w_{S}w · μ. We also write F_{w} for the simple finite-dimensional 𝔪-module of highest weight w_{S}w · μ.

3. Definition of Kostant modules

3.1. In this section we define Kostant modules in blocks of category O_{S} as simple modules having 𝔲-cohomology H^{∗}(𝔲, E) which as an 𝔪-module is multiplicity free and described by some poset ^{S′}W′.

3.2. Graded posets. Recall that a finite poset is called graded if it is an interval (i.e., it has a least and a greatest element) and if all maximal chains between any two elements x < y have the same length. On a graded interval [v, w] we then have a well-defined rank function given by r(x)=length of any maximal chain from v to x. We will write [v, w]_{j} for the set of all x in [v, w] for which r(x) = j.

It is well-known, and follows from a result of Deodhar [Deo1], that the posets of the form SW are graded and that the rank function on ^{S}W is the restriction of the length function l on W. However, the posets of the form ^{S}W^{J} are not graded in general.

Definition. For w ∈ ^{S}W^{J}, we say that L_{w} is a weak Kostant module in O_{S}^{μ} if there exists a graded interval [v, w] of ^{S}W^{J} such that as an 𝔪-module,

H^{i}(𝔲, L_{w}) ≃ ⨁_{x∈[v,w]_{r(w)−i}} F_{x},

where r is the rank function on [v, w].

We say that a weak Kostant module L_{w} is a Kostant module in O_{S}^{μ} if the graded interval [v, w] is isomorphic as a graded poset to a poset of the form ^{S′}W′ (for some Weyl group W′ and subset S′ of its set of simple roots).

Remark. Every finite-dimensional simple module is a Kostant module by Kostant’s theorem. Note that in our notation, a simple module L_{w} in O_{S}^{μ} is finite-dimensional if and only if μ + ρ is regular and w is the greatest element of ^{S}W^{J} = ^{S}W. The interval in this case is [v, w] = [e, w] = ^{S}W, and the rank function is the length function.

3.3. Kostant modules and Kazdan-Lusztig-Vogan polynomials. The definition of weak Kostant modules simplifies for regular blocks. Let O_{S}^{μ} = O_{S}^{reg} be any regular block; that is, μ + ρ is regular. For x, w ∈ ^{S}W define the relative Kazdan-Lusztig-Vogan polynomial ^{S}P_{x,w} by

SP_{x,w} = ∑_{i≥0} q^{\frac{l(w)−l(x)−i}{2}} dim Ext_{O_{S}}^{i}(N_{x}, L_{w}).

It follows from the Kazhdan-Lusztig conjectures (which are theorems in this setting; cf.[CC]) that the polynomial ^{S}P_{x,w} is equal to the parabolic Kazhdan-Lusztig polynomial that was defined by Deodhar [Deo2]. It is well known (and follows, for example, by a straightforward extension of the proof of [Sch, Lemma 5.13], the corresponding statement in ordinary category O), that

Hom_{𝔪}(F_{x}, H^{i}(𝔲, L_{w})) ≃ Ext_{O_{S}}^{i}(N_{x}, L_{w}) as vector spaces.

It then follows almost immediately from the definition of a weak Kostant module in Definition 3.2 (and the fact that ^{S}P_{e,w} ≠ 0) that the simple module L_{w} in O_{S}^{reg} is a weak Kostant module if and only if ^{S}P_{x,w} = 1 for every x ∈ [e, w].

3.4. Schubert varieties. Let G be a connected complex algebraic group with Lie algebra 𝔤 and let T ⊂ B ⊂ P ⊂ G be the connected closed subgroups corresponding to 𝔥 ⊂ 𝔟 ⊂ 𝔭 ⊂ 𝔤. The maximal torus T acts with finitely many fixed points on Y = G/P and these fixed points are naturally parametrized by the parabolic poset ^{S}W. The B-orbits in Y are the orbits of the T -fixed points. The closure of the orbit through the T -fixed point corresponding to w ∈ ^{S}W is denoted Y_{w} and is called a generalized Schubert variety. It follows from the work of Kazhdan-Lusztig [KL] and Deodhar [Deo2] that Y_{w} is rationally smooth if and only if SP_{x,w} = 1 for every x ∈ [e, w].

Thus we have a canonical 1-1 correspondence

{weak Kostant modules in O_{S}^{reg}} ↔ {rationally smooth Schubert varieties in Y },

where L_{w} corresponds to Y_{w}. To classify the weak Kostant modules in regular blocks we can use any of the many known tests for rational smoothness of Schubert varieties (cf. Billey-Lakshmibai [BL] for an overview). A test that will be useful later is the following, due to Carrell-Peterson [Car]. For w ∈ ^{S}W, define the Poincaré polynomial P_{w}(t) by

P_{w}(t) = ∑_{v≤w} t^{l(v)}.

Then Y_{w} is rationally smooth if and only if P_{w}(t) is palindromic.

3.5. Example. Consider (Φ, Φ_{S}) of type (D_{4}, A_{2}). Figure 2 shows the Hasse diagram of the poset ^{S}W. The nodes of the Hasse diagram enclosed in solid circles correspond to Kostant modules. The dashed circles correspond to weak Kostant modules, or equivalently, smooth Schubert varieties. (In simply-laced types, a result of Peterson says that a Schubert variety is rationally smooth if and only if it is smooth [CK].) The reader may want to verify that the Poincaré polynomials are palindromic. The nodes w enclosed by dashed circles are ones for which the Poincaré polynomial is palindromic, but the truncated poset [e, w] is not isomorphic to a poset of the form ^{S′}W′.

4. Standard Kostant Modules in Regular Blocks

4.1. In Section 5, we will classify the weak Kostant modules in regular blocks for any maximal parabolic in simply laced type in terms of subdiagrams of the Dynkin diagram of the pair. In this section, we explain, for a general parabolic, how to associate Kostant modules to subdiagrams. The Kostant modules that arise in this way we will call standard Kostant modules. In Section 5 we will then show that for maximal parabolics in simply laced type, every weak Kostant module in a regular block is a standard Kostant module.

Figure 2. Weak Kostant modules in a regular block for (D_{4}, A_{2})

4.2. Standard Kostant modules. Fix a standard parabolic subalgebra 𝔭 = 𝔭_{S} ⊂ 𝔤 and let P ⊂ G be the parabolic subgroup corresponding to 𝔭. To any standard parabolic subalgebra 𝔮 = 𝔭_{I} ⊂ 𝔤 we can associate a smooth Schubert variety in G/P as follows. Let Q ⊂ G be the parabolic subgroup corresponding to 𝔮 and let Q = LU be the Levi decomposition of Q. Then L ∩ B is a Borel subgroup of the reductive group L and L ∩ P is a parabolic subgroup of L. Furthermore, B = (L ∩ B)U. Now consider the inclusion L/(L ∩ P) ↪ G/P, which in fact is a closed embedding of smooth projective varieties.

Lemma. Under the closed embedding L/(L∩P) ↪ G/P, the image of any (L∩B)-orbit in L/(L ∩ P) is a B-orbit in G/P. In particular, the image of L/(L ∩ P) is a smooth Schubert variety in G/P.

Proof. Let x ∈ L and b ∈ B. Since B = (L ∩ B)U, we may write b = b_{L}u with b_{L} ∈ L ∩ B and u ∈ U. Then, since U is normalized by L, we have bx = b_{L}ux = b_{L}xu′ for some u′ ∈ U. Since U ⊂ P (in fact, U ⊂ B ⊂ P), it follows that the B-orbit of xP is the image of the (L ∩ B)-orbit of x(L ∩ P). _{□}

Remark. The claim of the lemma is true if we replace L by the commutator group L′ = [L, L], which is connected semisimple. As before, L′ ∩B is a Borel subgroup of L′ and L′ ∩P is a parabolic subgroup of L′. Furthermore, since L = Z(L)L′, where Z(L) is the center of L, it follows that the inclusion L′ ⊂ L induces an isomorphism L′/(L′ ∩ P) = L/(L ∩ P).

Let Y_{w} be the Schubert variety that is the image of L/(L ∩ P) under the embedding L/(L ∩ P) ↪ G/P. In light of 3.4, there is then a weak Kostant module L_{w} associated to Y_{w}. Note that L_{w} is in fact a Kostant module because [e, w] ≃ ^{S′}W′ = ^{S∩I}W_{I}, where W′ = W_{I} is the Weyl group of L and S′ = S ∩ I ⊂ I is the subset of the simple roots of L that defines the parabolic subgroup L ∩ P ⊂ L. The element w is the maximal element of ^{S∩I}W_{I}, viewed as an element of ^{S}W. In the following, to emphasize the dependence on I, we will write w = φ(I). The Kostant modules in O_{S}^{reg} of the form L_{φ(I)} we will call standard Kostant modules.

4.3. Subdiagrams. Let D be the Dynkin diagram of the pair (𝔤, 𝔭). The Dynkin diagram of the pair (𝔩, 𝔩∩𝔭), where 𝔩 is the Levi subalgebra of 𝔮 = 𝔭_{I}, may be viewed as a subdiagram of D. More precisely, let D(I) denote the subdiagram of D with nodes corresponding to the simple roots in I and the crossed nodes corresponding to the simple roots in I \ S. Then D(I) is the diagram of the pair (𝔩, 𝔩 ∩ 𝔭). Clearly, any subdiagram of D is of the form D(I). Furthermore, the subdiagram D(I) determines the set I. Thus we get a map from the set of subdiagrams of D to the set of Kostant modules in O_{S}^{reg} given by D(I) ↦ L_{φ(I)}. In the following we will distinguish a certain subset of the set of subdiagrams of D, such that the map restricted to this subset is one-to-one.

Suppose that I ≠ ∅ and let I = I_{1} ∪ ⋯ ∪ I_{r} be the partition of I corresponding to the decomposition D(I) = D(I_{1}) ∪ ⋯ ∪ D(I_{r}) of the diagram D(I) into connected components. Then φ(I) = φ(I_{1}) ⋯ φ(I_{r}). We say that D(I_{j}) is an S-trivial component of D(I) if φ(I_{j}) = e. Clearly, D(I_{j}) is an S-trivial component of D(I) if and only if I_{j} ⊂ S, i.e., if and only if D(I_{j}) does not contain any crossed nodes. If I = ∅ we let D(I) be the empty subdiagram of D, which (vacuuously) has no S-trivial components. With this terminology in place, we have the following result.

Proposition. We have an injection

{subdiagrams of D with no S-trivial components} ↪ {Kostant modules in O_{S}^{reg}}

given by D(I) ↦ L_{φ(I)}.

Remark. If S^{c} = {α}, i.e., if 𝔭 is a maximal parabolic subalgebra of 𝔤, then the non-empty subdiagrams of D with no S-trivial components are the connected subdiagrams containing the node α. For simplicity of stating results in this setting, we shall regard the empty subdiagram of D as belonging to the set of connected subdiagrams containing α.

The key in the proof of the proposition is the following lemma. We assume that the lemma is well known to experts, but since we couldn’t find a proof in the literature, we provide a proof.

Lemma. Suppose that S ≠ Δ and let ^{S}w be the maximal element of ^{S}W. Then any reduced expression for ^{S}w involves all simple reflections.

Proof. Fix α_{p} ∈ Δ \ S. For the purposes of this proof only, number the simple roots α_{1}, …, α_{p}, …, α_{m}, α_{m+1}, …, α_{n} where α_{1}, …, α_{m} is a maximal chain containing α_{p} in the Dynkin diagram, and α_{m+1}, …, α_{n} is the other branch (if any) of the diagram, so that α_{1}, α_{m}, and α_{n} (if m < n) are the end nodes of the diagram. Consider the Coxeter element

w := s_{p}s_{p−1} … s_{1}s_{p+1}s_{p+2} … s_{m}s_{m+1} … s_{n}.

We claim that the given expression for w is reduced; every reduced expression for w begins with s_{p} and involves every simple reflection exactly once; and w ∈ ^{S}W.

First, any product of distinct simple reflections is necessarily reduced (cf. [Hum, Exercise 1.13]). If not, one could find an initial expression s_{i_{1}} … s_{i_{k−1}}s_{i_{k}} = xs_{i_{k}} with l(xs_{i_{k}}) < l(x); i.e., x(α_{i_{k}}) < 0. On the other hand, by the formula for a reflection, the coefficient of α_{i_{k}} in x(α_{i_{k}}), when written as a linear combination of simple roots, is 1 (since α_{i_{1}}, …, α_{i_{k−1}} are distinct from α_{i_{k}}). This contradiction proves the first claim.

Since the set of simple reflections appearing in a fixed reduced expression for any element x ∈ W depends only on x and not on the reduced expression [Hum, Corollary 5.8(b)], every reduced expression for w involves every simple reflection (and exactly once, by length considerations).

Any other reduced expression for a Coxeter element w can be obtained from the given one by applying the “commuting relations” s_{i}s_{j} = s_{j}s_{i} [Bou, Chap. 4, Sec. 1, Prop. 5 and Exercise 13]. If j ≠ p, then there is an s_{i} occuring to the left of s_{j} in the given expression for w with s_{i}s_{j} ≠ s_{j}s_{i}. This means that s_{i} will occur to the left of s_{j} in every reduced expression for w, and in particular, no reduced expression can begin with s_{j}. By the characterization of ^{S}W as those elements of W all of whose reduced expressions begin with a reflection corresponding to a simple root not in S, it follows that w ∈ ^{S}W. This proves the claim.

Since ^{S}w is the greatest element in the Bruhat order of ^{S}W, we have w ≤ ^{S}w. By Deodhar’s subexpression condition, some reduced expression for w occurs as a subexpression of any given reduced expression for ^{S}w. By the claim, this proves the lemma. _{□}

Proof of Proposition. Let D(I) = D(I_{1}) ∪ ⋯ ∪ D(I_{r}) be the decomposition of the (nonempty) subdiagram D(I) of D into connected components and assume that none of the D(I_{j}) is S-trivial. Then, by definition, S ∩ I_{j} ≠ I_{j} for all j. Recall that φ(I_{j}) is the maximal element of ^{S∩I_{j}}W_{I_{j}}. By the lemma above, any reduced expression for φ(I_{j}) involves all simple reflections s_{i}, i ∈ I_{j}. Since I = I_{1} ∪ ⋯ ∪ I_{r} and I_{j} ⊥ I_{j′} for j ≠ j′, this implies that any reduced expression for φ(I) = φ(I_{1}) ⋯ φ(I_{r}) involves all simple reflections s_{i}, i ∈ I. Thus, I is uniquely determined by φ(I) and the proposition follows. _{□}

4.4. Example. Consider (Φ, Φ_{S}) = (F_{4}, C_{3}). Figure 3 shows the Hasse diagram of ^{S}W with the nodes corresponding to weak Kostant modules circled as before. There are a total of eight weak Kostant modules, all of which are Kostant modules. Only five of these eight Kostant modules are standard. The subdiagrams D(I) corresponding to I = ∅, {a}, {a, b}, {a, b, c}, and Δ = {a, b, c, d} are drawn next to the corresponding nodes of standard Kostant modules.

4.5. Enright-Shelton equivalences. We conclude this section by giving an interpretation of standard Kostant modules via certain equivalences of categories that were introduced by Enright and Shelton in [ES1]. Define the truncated category O_{t}(𝔤, 𝔭, φ(I)) as the full subcategory of O(𝔤, 𝔭, reg) = O_{S}^{reg} consisting of all modules V such that [V : L_{w}] ≠ 0 implies

a b c d

a b c b d

a d b

d a c

c a b

b a

c a b

d a c

a d b

b d c b a

∅

Figure 3. Kostant modules in a regular block for (F_{4}, C_{3})

w ≤ φ(I). The simple modules in O_{t}(𝔤, 𝔭, φ(I)) are then the modules L_{w} with w ≤ φ(I). Let 𝔩′ = [𝔩, 𝔩], where 𝔩 is the Levi subalgebra of 𝔮 = 𝔭_{I} as in Section 4.3. In [ES1], Enright and Shelton proved (see also [Soe]) the following result.

Proposition. There is an equivalence of categories between O(𝔩′, 𝔩′∩𝔭, reg) and O_{t}(𝔤, 𝔭, φ(I)), sending the generalized Verma module N′_{w} to N_{w} and the simple module L′_{w} to L_{w} for all w ∈ ^{S′}W′ ⊂ ^{S}W.

Note that according to our definition of O(𝔩′, 𝔩′ ∩ 𝔭, reg), the simple module in O(𝔩′, 𝔩′ ∩ 𝔭, reg) corresponding to the maximal element of ^{S′}W′ is finite dimensional.

Corollary. Every standard Kostant module in O_{S}^{reg} admits a Bernstein-Gelfand-Gelfand (BGG) resolution in terms of generalized Verma modules.

Proof. By a result of Lepowsky [Lep], every simple finite dimensional 𝔤-module admits a BGG resolution in O_{S} . By the remarks above every standard Kostant module then admits a BGG resolution as well. _{□}

5. Regular Blocks for Maximal Parabolics in Simply Laced Type

5.1. In this section we assume that 𝔤 is simply-laced, and that S^{c} = {α} (i.e. that 𝔭_{S} is a maximal parabolic subalgebra of 𝔤). We work in a fixed regular block O_{S}^{reg}. Under these assumptions, we prove that there are bijections between the following sets: {connected subdiagrams of D containing α} (recall our convention that we consider the empty subdiagram to belong to this set, since we really mean the set of subdiagrams with no S-trivial component); {Kostant modules in O_{S}^{reg}}; and {weak Kostant modules in O_{S}^{reg}}.

We outline the idea of the proof. According to Proposition 4.3, we have an injection of the first set into the second, and by definition the second is a subset of the third. Recall that L_{w} is weak Kostant (if and) only if the Poincaré polynomial P_{w}(t) is palindromic. Set Pal = { w ∈ ^{S}W | P_{w}(t) is palindromic }. We will show that

(5.1.1)  |Pal| ≤ #{connected subdiagrams of D containing α}.

This will clearly prove that all four sets have the same cardinality.

Theorem. Let 𝔤 be a simply-laced simple Lie algebra, and let S ⊂ Δ with S^{c} = {α}. Let D be the corresponding Dynkin diagram with one crossed node. Fix a regular block O_{S}^{reg} of category O_{S}. Then every weak Kostant module in O_{S}^{reg} is a Kostant module, and there is a bijection

\{ Kostant modules in O_{S}^{reg} \} ↔ \{ \begin{matrix}connected subdiagrams \\ of D containing α \end{matrix} \}

(where we declare the empty subdiagram to belong to the set on the right).

Proof. It suffices to prove (5.1.1). Certainly the identity element e ∈ Pal. Fix w ∈ ^{S}W with l(w) > 0. As s_{α} is the unique element of ^{S}W of length 1, and s_{α} ≤ w, we have

(5.1.2)  P_{w}(t) = 1 + t + ⋯ + ct^{l(w)−1} + t^{l(w)} for some c ∈ ℕ.

A necessary condition for w ∈ Pal is c = 1. In particular, if w ∈ Pal then

(5.1.3)  there is at most one simple reflection s such that w > ws ∈ ^{S}W.

In the classical types, we use Proctor’s parametrization [Pro] for ^{S}W. (For reasons of personal preference we reverse his strings, left-to-right, and also reverse the Bruhat order, so that e is the least element.) Assume α = α_{r} in the Bourbaki ordering of simple roots.

First suppose we are in type A_{n−1}. Then^{S}W consists of all permutations w of e = 1^{r}0^{n−r}. The order-generating relations are given by

(5.1.4)  S_{ij}(w) < w if 1 ≤ i < j ≤ n and w_{i} < w_{j},

where S_{ij} acts on the n-tuple w by interchanging w_{i} and w_{j}. Here S_{ij} is the simple reflection s_{i} when j = i + 1. Evidently the elements satisfying (5.1.3) are precisely those of the form

(5.1.5)  w = 1 … 1^{k} 0 … 0 1 … 1^{l} 0 … 0, 0 ≤ k < r < l ≤ n or k = r = l.

On the other hand the connected subdiagrams of D containing α are obtained by “pruning” (deleting) nodes k and l from D, where 0 ≤ k < r < l ≤ n or k = r = l, and then selecting the connected component containing node r in the remaining graph. (Here, the case k = r = l produces the empty graph, which corresponds to the Kostant module L_{e}.) Since these two sets are parametrized by the identical values of k and l, they have the same cardinality. (In fact, it is easy to see that the element in (5.1.5) is the maximal element of the subposet of ^{S}W corresponding to the connected sub-Dynkin diagram obtained by pruning nodes k and l as described.) This proves the desired bijection in type A.

Next suppose we are in type D_{n}. Assume first that r < n − 1. Then ^{S}W consists of all signed permutations of e = 1^{r}0^{n−r}. We denote −1 by ¯1 when depicting signed permutations. The order-generating relations are given by S_{ij} as in (5.1.4), together with

(5.1.6)  SN_{ij}(w) < w if 1 ≤ i < j ≤ n and w_{i} + w_{j} < 0,

where SN_{ij} acts by interchanging and negating w_{i} and w_{j}. Here SN_{n−1,n} is the simple reflection s_{n}. The elements satisfying (5.1.3) are of the following types:

(5.1.7)  \begin{aligned}w^{1} & = 1 … 1^{k} 0 … 0 1 … 1^{l} 0 … 0, 0 ≤ k < r < l ≤ n or k = r = l, \\ w^{2} & = 1 … 1^{k} 0 … 0 1 … 1¯1, 0 ≤ k < r, \\ w^{3} & = 1 … 1^{k} ¯1 … ¯1 0 … 0, 0 ≤ k < r, \\ w^{4} & = 1 … 1^{k} 0 … 0^{l} ¯1 … ¯1^{m} 1 … 1^{p} 0 … 0, 0 ≤ k ≤ l < m < p ≤ n \\ w^{5} & = 1 … 1^{k} 0 … 0 ¯1^{l} … ¯1¯1, 0 ≤ k ≤ r − 2, \\ w^{6} & = 1 … 1^{k} 0 … 0 ¯1 … ¯1^{l} 0 … 0, 0 ≤ k < r < l ≤ n − 2. \end{aligned}

The element w^{1} is the maximal element of the sub-poset of ^{S}W corresponding to the subdiagram of D consisting of nodes k + 1 through l − 1 (or the empty subdiagram, if k = r = l). The element w^{2} corresponds to the subdiagram consisting of nodes k + 1 through n − 2 and n. The element w^{3} corresponds to the subdiagram consisting of nodes k + 1 through n. These exhaust all the possible connected subdiagrams of D containing node r. Thus it remains to show that elements of the form w^{4}, w^{5}, w^{6} do not parametrize weak Kostant modules. That is, we must show that their Poincaré polynomials are not palindromic.

Let w = w^{4}. Then ws_{m} → w. But also, by [Pro, Cor. 5D], S_{l,m+1}(w) → w if k < l, and S_{m,p+1}(w) → w if p < n. At least one of these conditions holds since r < n − 1. Thus, in the notation of (5.1.2), c ≥ 2 and P_{w}(t) is not palindromic.

Let w = w^{5}. Note that the condition on k implies l ≤ n − 1; the case l = n is of type w^{2}. Then ws_{n} → w. But also, by [Pro, Cor. 5D], SN_{l−1,n}(w) → w. Again, P_{w}(t) is not palindromic.

Let w = w^{6}. This is the most challenging case, because we must go several levels down from w (and up from e) to detect the failure of palindromicity. As a warm-up,

consider the case r = 1, where w = 0 … 0 ¯1^{l} 0 … 0 with 1 < l ≤ n − 2. Write [e, w]_{i} = { x ≤ w | l(x) = i }. We claim that #[e, w]_{l(w)−i} = #[e, w]_{i} = 1 for 0 ≤ i < n − l but #[e, w]_{l(w)−(n−l)} = 2 while #[e, w]_{n−l} = 1, and thus P_{w}(t) is not palindromic. Indeed, for

i ≤ n − l, [e, w]_{i} = { 0 … 0 1^{i+1} 0 … 0 } and for i < n − l, [e, w]_{l(w)−i} = { 0 … 0 ¯1^{l+i} 0 … 0 },

but [e, w]_{l(w)−(n−l)} = { 0 … 0 ¯1 } ∪ { 0 … 0 1 }. Returning to general r < n − 1, we claim that #[e, w]_{l(w)−i} ≥ #[e, w]_{i} for 0 ≤ i < n − l but #[e, w]_{l(w)−(n−l)} > #[e, w]_{n−l}. In fact, any element x of [e, w]_{i} for 0 ≤ i ≤ n − l is obtained from e by moving some of the last r − k 1’s to the right a total of i positions. (Moving any of the first k 1’s would produce an element not dominated by w. And there are no non-simple coverings among the elements of length at most n − l.) To any such x there is an element x′ of [e, w]_{l(w)−i} obtained by moving the corresponding ¯1’s in w the same number of positions to the right. (The last 1 in e corresponds to the last ¯1 in w, and so on.) The map x ↦ x′ is clearly an injection.

On the other hand, the element 1 … 1^{k} 0 … 0 ¯1 … ¯1^{l−1} 0 … 0 1 ∈ [e, w]_{l(w)−(n−l)} is not in the image of this map. This shows that P_{w}(t) is not palindromic, and completes the proof for type D_{n} when r < n − 1.

The labeled posets ^{S}W for r = n − 1 and r = n are isomorphic, so it suffices to consider the case r = n. Then ^{S}W consists of all signed permutations of e = 1^{n} containing an even number of ¯1’s. The elements satisfying (5.1.3) are of the following types:

(5.1.8)  \begin{aligned}w^{1} & = 1 … 1 \\ w^{2} & = 1 … 1^{k} ¯1 1 … 1 ¯1, 0 ≤ k ≤ n − 2, \\ w^{3} & = 1 … 1^{k} ¯1 … ¯1 ±1, 0 ≤ k < n − 2, \\ w^{4} & = 1 … 1^{k} ¯1 … ¯1^{m} 1 … 1 ±1, 0 ≤ k < m − 1, m < n − 1, \end{aligned}

with ± chosen in w^{3} and w^{4} to make the total number of −1’s even. Then w^{1} corresponds to the empty subdiagram of D, w^{2} to the subdiagram containing nodes {k + 1, …, n} \ {n − 1}, and w^{3} to the subdiagram containing nodes {k + 1, …, n}. This accounts for all connected subdiagrams of D containing node n, so we must show that w = w^{4} has nonpalindromic Poincaré polynomial. Notice that ^{S}W contains a unique element of length 2, namely 1 … 1¯11¯1. However, there are two elements in [e, w]_{l(w)−2}, namely ws_{m}s_{m−1} and ws_{m}s_{m+1} (or ws_{m}s_{m+2} if w = 1 … 1¯1 … ¯11¯1). Thus the coefficients of t^{2} and t^{l(w)−2} differ in P_{w}(t).

Finally assume we are in type E_{n}. We wrote a computer program to calculate ^{S}W (using the characterization ^{S}W = { w ∈ W | w^{−1}(Φ_{S}^{+}) ⊂ Φ^{+} }) and then find the elements in ^{S}W having palindromic Poincaré polynomial. (More complicated computer analyses of rational smoothness for parabolic Schubert varieties in type E have been done by Billey-Postnikov [BiP].) The Bruhat order was determined via the following recursive algorithm. Given x, w ∈ ^{S}W with w ≠ e, find a simple reflection s such that w > ws =: w′. Then necessarily w′ ∈ ^{S}W, by [Deo1]. Set x′ equal to x or xs, whichever is smaller. Then x ≤ w if and only if x′ ≤ w′, by op. cit. It is easy to calculate by hand the number of connected subdiagrams of D containing node r, for each choice of r. In every case, we found the number of elements with palindromic Poincaré polynomial to be the same as the number of such connected subdiagrams. These numbers are listed in Table 1. _{□}

r E_{6} E_{7} E_{8}
1 9 11 13
2 11 14 17
3 15 19 23
4 19 25 31
5 15 22 29
6 9 17 25
7  10 19
8  11

Table 1. Number of (weak) Kostant modules for maximal parabolics

6. Arbitrary Blocks for Hermitian Symmetric Pairs

6.1. In this section we assume that (Φ, Φ_{S}) corresponds to a Hermitian symmetric pair. We prove that in any (nonempty) integral block O_{S}^{μ}, every weak Kostant module is a Kostant module, and these are parametrized by a set of the form {connected subdiagrams of D′ containing α′}, for a certain Dynkin diagram D′ with a single crossed node α′. The key tool is an equivalence of categories due to Enright (based on work of Enright and Shelton).

As remarked in the introduction, the (weak) Kostant modules in regular (integral) Hermitian symmetric categories were first worked out by Collingwood [Col]. We give new proofs of his results.

6.2. The regular case. Let (Φ, Φ_{S}) correspond to a Hermitian symmetric pair. Then S is a maximal proper subset of Δ; put S^{c} = {α} as before. Assume in this subsection that μ is regular.

We need some additional notation in case D is not simply laced; i.e., when D = (B_{n}, B_{n−1}) (resp. (C_{n}, A_{n−1})). Let D^{∨} be the Dynkin diagram dual to D, and let α^{∨} be the simple root in D^{∨} dual to α. (The fundamental dominant weight associated to α is cominuscule, while that associated to α^{∨} is minuscule.) Let Φ^{∨} be the root system of D^{∨} with simple roots Δ^{∨}. Let S^{∨} = Δ^{∨} \ {α^{∨}}. Define the simply-laced cover of D^{∨} to be the Dynkin diagram D̃^{∨} of type A_{2n−1} (resp. D_{n+1}). It can be useful to imagine D̃ “folding” at node n (resp. n − 1) to “cover” D, as depicted in Figure 4. Let Φ̃^{∨} be the root system of D̃^{∨} with simple roots Δ̃^{∨}. Let S̃^{∨} = Δ̃^{∨} \ {α̃^{∨}}, where α̃^{∨} = α_{1} (resp. α_{n+1}). Thus we have associated to D = (B_{n}, B_{n−1}) (resp. (C_{n}, A_{n−1})) a simply-laced Hermitian symmetric pair D̃^{∨} = (A_{2n−1}, A_{2n−2}) (resp. (D_{n+1}, A_{n})).

2n − 1  n 1_{1  n}  ^{1  n n + 1}_{1  n}

Figure 4. Simply-laced covers

Theorem. Let O_{S}^{reg} be a regular block for a Hermitian symmetric pair. Then every weak Kostant module is a Kostant module. Moreover there is a Dynkin diagram D′ with one crossed node α′, such that

\{ Kostant modules in O_{S}^{reg} \} ↔ \{ \begin{matrix}connected subdiagrams \\ of D′ containing α′ \end{matrix} \}

(where we declare the empty subdiagram to belong to the set on the right). If D is simplylaced, then D′ = D and α′ = α; otherwise, D′ = D̃^{∨} and α′ = α̃^{∨}.

Proof. By Theorem 5.1, if D is simply-laced, there is nothing to prove. If not, let D^{∨}, D̃^{∨}, etc. be as above. Since the Weyl groups for B_{n} and C_{n} are identical, so are the labeled posets SW associated to D and D^{∨} (with crossed nodes α and α^{∨}, respectively), and they have the same KL polynomials and hence the same parameters for the (weak) Kostant modules. Thus we may replace D with D^{∨}. Now let G^{∨} be a connected simple algebraic group over ℂ having root system Φ^{∨}, P^{∨} ⊂ G^{∨} a parabolic subgroup corresponding to S^{∨}, and similarly for G̃^{∨} and P̃^{∨}. The flag varieties G^{∨}/P^{∨} and G̃^{∨}/P̃^{∨} are isomorphic, via an isomorphism (coming from the embeddings Sp(2n, ℂ) ↪ SL(2n, ℂ) and SO(2n + 1, ℂ) ↪ SO(2n + 2, ℂ)) which identifies Schubert varieties in the two spaces [BrP, Sec. 3.1]. Thus, by our identification of weak Kostant modules with rationally smooth Schubert varieties, and Theorem 5.1, we have that the Kostant and weak Kostant modules are both parametrized by connected subdiagrams of D̃^{∨} containing α̃^{∨} (together with the empty diagram). _{□}

6.3. The singular case. Maintain the assumptions on S from the previous subsection, but now assume that μ is singular. Define J as in Section 2.3. If J contains two simple roots adjacent in D, then O_{S}^{μ} is empty, by [BN, Cor. 4.2] or [Enr, Lem. 3.1a]. So we may assume that Φ_{J} is of type A_{1}×⋯×A_{1} (t factors, for some t ∈ ℕ). Suppose we are in one of the three “complicated” Hermitian symmetric cases, (A_{n}, A_{r−1} × A_{n−r}), (C_{n}, A_{n−1}), or (D_{n}, A_{n−1}). In 1987 Enright and Shelton [ES1] proved in the semiregular case (t = 1) that the category O(Φ, Φ_{S}, Φ_{J}) is equivalent to a regular block of a Hermitian symmetric category of rank n − 2 (or, in one case, a direct sum of two copies of such a block). The following year Enright [Enr] realized that essentially the same proof handles the case of general J, with the rank of the regular block being n − 2t. As mentioned above, these equivalences are key to understanding the (weak) Kostant modules in these three cases.

To get a feel for how the equivalences work, it may be helpful to look at an example. Consider type A_{n−1} with S^{c} = {r} and J = {j_{1}, …, j_{t}} (viewing S, J ⊂ {1, …, n − 1} in the obvious way). After applying a translation functor, and shifting by a constant vector to make all entries non-negative, we may assume that μ + ρ = (a_{1}, …, a_{n}) (in the ε_{i}-coordinates) with a_{1} = 0, a_{i+1} = a_{i} + 1 if i ∉ J, and a_{i+1} = a_{i} if i ∈ J. The permutations of μ + ρ which are highest weights plus ρ of simple modules in O(Φ, Φ_{S}, Φ_{J}) have their first r entries strictly decreasing, and likewise their last n − r entries strictly decreasing. This implies that, of each matched pair of equal entries a_{j} = a_{j+1} (j ∈ J), one entry must occur in the first r coordinates and the other in the last n − r. Now these matched pairs can all be deleted; this operation is clearly invertible. The resulting (n − 2t)- tuples correspond to the weights for a regular block of the Hermitian symmetric category O(A_{n−1−2t}, A_{r−1−t} × A_{n−1−r−t}). Enright’s result is that this bijection on weights arises from an actual equivalence of categories. Similar weight analyses can be easily made for the other two “complicated” cases.

Theorem. Let (Φ, Φ_{S}) correspond to a Hermitian symmetric pair. In case (Φ, Φ_{S}) = (C_{n}, A_{n−1}), assume J does not contain the long simple root. Then whenever O(Φ, Φ_{S}, Φ_{J}) is a nonempty singular block, every weak Kostant module is a Kostant module. Moreover, there is a Dynkin diagram D′ with a single crossed node α′ such that the set of Kostant modules in O(Φ, Φ_{S}, Φ_{J}) is in bijection with one or two copies of {connected subdiagrams of D′ containing α′}. The pairs (D′, α′), and the number of copies are given explicitly in Table 2. (The case D′ = ∅ corresponds to the block O(∅, ∅) having only one simple module.)

D  α  |J|  D′  α′  # Copies
(A_{n}, A_{r−1} × A_{n−r}) α_{r}  t  (A_{n−2t}, A_{r−t−1} × A_{n−r−t}) α_{r−t}  1
(B_{n}, B_{n−1})  α_{1}  1 (short)  ∅  –  1
(B_{n}, B_{n−1})  α_{1}  1 (long)  ∅  –  2
(C_{n}, A_{n−1})  α_{n} t (all short)  (D_{n+1−2t}, A_{n−2t})  α_{n+1−2t}  1
(C_{n}, A_{n−1})  α_{n} t (1 long)  (D_{n+1−2t}, A_{n−2t})  α_{n+1−2t}  2
(D_{n}, D_{n−1})  α_{1}  1  (A_{1}, ∅)  α_{1}  1
(D_{n}, D_{n−1})  α_{1} 2 ({n − 1, n})  ∅  –  1
(D_{n}, A_{n−1})  α_{n}  t  (D_{n−2t}, A_{n−2t−1})  α_{n−2t}  1
(E_{6}, D_{5})  α_{6}  1  (A_{5}, A_{4})  α_{5}  1
(E_{6}, D_{5})  α_{6}  2  ∅  –  1
(E_{7}, E_{6})  α_{7}  1  (D_{6}, D_{5})  α_{1}  1
(E_{7}, E_{6})  α_{7}  2  (A_{1}, ∅)  α_{1}  1
(E_{7}, E_{6})  α_{7} 3 ({2, 5, 7})  ∅  –  1

Table 2. Data for singular Hermitian symmetric categories Remark. The theorem should “morally” be true for the excluded case O(C_{n}, A_{n−1}, Φ_{J}) where J contains the long simple root. When such a category is nonempty, it splits into a direct sum of two blocks, each equivalent to O(C_{n−2|J|}, A_{n−2|J|−1|}, ∅), by [Enr, Prop. 3.2(b)]. So a simple module L_{w} in the original category ought to be a Kostant module if it corresponds to a Kostant module in either of the direct summands. Unfortunately, the relevant interval [v, w] in ^{S}W^{J} will usually include parameters x from the other direct summand, which do not appear in the cohomology of L_{w}. Thus L_{w} cannot satisfy Definition 3.2 of weak Kostant modules. In Section 7.3 below we propose a new ordering on^{S}W^{J} which will rectify this problem and make L_{w} a Kostant module; see Section 7.4 for details. In anticipation of that discussion, we have included the data for this case in Table 2.

Proof. First assume we are in type (A_{n}, A_{r−1} × A_{n−r}), (C_{n}, A_{n−1}), or (D_{n}, A_{n−1}). Then by [Enr, Prop. 3.2(a)] there is an equivalence of categories O(Φ, Φ_{S}, Φ_{J}) ≃ O(Φ′, Φ′_{S′}, ∅), where (Φ′, Φ′_{S′}) corresponds to a Hermitian symmetric pair of the same type but of rank n − 2|J|. The equivalence induces a bijection of partially ordered sets ^{S}W^{J} ↔ ^{S′}W′, and identifies correspondingly-parametrized simple modules, GVMs, etc. In particular, the equivalence matches up weak Kostant modules (resp. Kostant modules) in the two categories.

Next assume we are in one of the remaining classical types, (B_{n}, B_{n−1}) or (D_{n}, D_{n−1}). The structure of the semiregular categories (J = {j}) was described in [BN, Sec. 4.1]. In type B if j < n there are two simple modules in two separate blocks, while if j = n there is just one simple module. In type D for any j there are two simple modules having an extension between them. Since the poset ^{S}W in type B is a chain, the set ^{S}W^{J} is empty whenever |J| ≥ 2, so the corresponding categories are all empty. In type D, ^{S}W^{J} is a “chain with a diamond in the middle,” the edges up from the bottom of the diamond being labeled by n − 1 and n. So for |J| ≥ 2 the category is empty unless J = {n − 1, n}, in which case it contains a single simple module. The theorem follows from these analyses.

Finally assume we are in one of the exceptional cases, (E_{6}, D_{5}) or (E_{7}, E_{6}). The structure of the singular blocks of these categories was analyzed in [ES2]. Begin with (E_{6}, D_{5}). The semiregular categories are all equivalent, and have the same structure as the regular category for (A_{5}, A_{4}): the poset of highest weights is a chain of length six, and if these are numbered 1 through 6 with V_{1} the irreducible GVM, then dim Ext^{k}(V_{i}, L_{j}) = δ_{k,j−i} [ES2, Prop. 2.3]. In particular, every simple module is a Kostant module, as for O(A_{5}, A_{4}, ∅). If J consists of any two orthogonal simple roots, it is easy to see by inspection of [ES2, Fig. 3.1] that SW^{J} contains only a single node, so the block consists of a single simple module. Now consider (E_{7}, E_{6}). The semiregular categories are again all equivalent, and have the same Hasse diagram and Ext(V_{x}, L_{w})-structure as O(D_{6}, D_{5}, ∅) [ES2, Prop. 3.9]. In particular, the (weak) Kostant modules are the same as for the regular (D_{6}, D_{5}) category. Similarly the categories in which J consists of two orthogonal simple roots are all equivalent, and have the same structure as O(A_{1}, ∅, ∅) [ES2, Lemma 3.5]. Lastly, if J consists of three orthogonal simples, then by inspection of [ES2, Fig. 3.1], the category is nonempty only if J = {2, 5, 7} (in the Bourbaki labeling), in which case it contains only a single simple module. _{□} 6.4. A unified approach. In this subsection we present a unified construction of the diagram D′ associated to a Hermitian symmetric diagram D with a single crossed node α in Theorem 6.3. Recall our convention that if all roots are the same length, they are short.

Let {γ_{1}, …, γ_{u}} be a maximal set of strongly orthogonal short roots in Φ(𝔲), defined as follows. Let γ_{1} be the highest short root in Φ(𝔲), and for i > 1 let γ_{i} be the highest short root in Φ(𝔲) that is orthogonal to γ_{1}, …, γ_{i−1}. (These are strongly orthogonal because 𝔲 is abelian, so the sum of two roots in Φ(𝔲) is never a root.) For 0 ≤ t ≤ u define Φ^{(t)} = { α ∈ Φ | (α, γ_{i}) = 0 for 1 ≤ i ≤ t }. We construct a sequence of Dynkin diagrams D^{(t)} associated to the root systems Φ^{(t)} as follows. Starting with D^{(0)} = D, create an extended Dynkin diagram by attaching an extra node corresponding to −γ_{1}. Then delete −γ_{1} and all nodes adjacent to it, to obtain D^{(1)}. In general, D^{(t)} is obtained from D^{(t−1)} by the same process (attaching −γ_{t}). One can check that D^{(t)} again corresponds to a Hermitian symmetric pair. (The Dynkin diagram D^{(t)} may be disconnected, so that O(Φ^{(t)}, Φ^{(t)} ∩ Φ_{S}) is a direct product of the categories corresponding to its connected components, all but one of which will be trivial. We may therefore delete these S-trivial components, replacing D^{(t)} with its connected component containing α. In two cases, (A_{n}, A_{n−1}) and (B_{n}, B_{n−1}), the simple root α is adjacent to −γ_{1}, so there is no component of D^{(1)} containing α. In this situation we replace D^{(1)} with ∅, with the same meaning as explained in the statement of Theorem 6.3. We do the same thing in general if α ∉ D^{(t)}.) In fact, comparison of the (connected) diagrams so obtained with those in Table 2 yields the following unified statement.

Theorem. Let (Φ, Φ_{S}) correspond to a Hermitian symmetric pair. If O(Φ, Φ_{S}, Φ_{J}) is a nonempty singular block with |J| = t, then the diagram D′ of Theorem 6.3 is D^{(t)} if D^{(t)} is simply laced, otherwise the simply connected cover of its dual.

Example. Let D = (E_{7}, E_{6}) and let O_{S}^{μ} be a semi-regular block; i.e., the rank of the singular root system is t = 1. The theorem says that the Kostant modules in O_{S}^{μ} are parametrized by the subdiagrams of D^{(1)} = (D_{6}, D_{5}), since in the extended Dynkin diagram of E_{7}, −γ_{1} is attached to the first simple root (see Figure 5). To describe explicitly the correspondence of the theorem, consider for example the case J = {d}. In the Hasse diagram of ^{S}W (left side of Figure 5) a node corresponds to an element of ^{S}W^{J} if it has an edge with label d going up (these edges are indicated in bold). The right side of Figure 5 shows the Hasse diagram of ^{S}W^{J}, with dashed edges indicating coverings of length difference greater than one. This poset is isomorphic to the parabolic poset ^{S′}W′ corresponding to D^{(1)}. The subdiagrams of D^{(1)} corresponding to the Kostant modules are listed along the right side of the figure.

7. Singular Blocks in the General Case

7.1. Recall from our discussion in Section 6 that for Hermitian symmetric pairs, every singular block is equivalent to a regular block (or a direct sum of two regular blocks) for some other Hermitian symmetric pair via equivalences of categories due to Enright and Shelton. However, in the non-Hermitian symmetric setting, singular blocks are more complicated than regular blocks. In this section we briefly explain how, at least in principle, one can decide whether a simple module in a singular block is a weak Kostant module via Kazhdan-Lusztig polynomials. We then introduce a new ordering on the set ^{S}W^{J} that is equal to

d d d d d d d d d d d d −γ_{1}  d  ∅

Figure 5. Kostant modules in a semi-regular block for (E_{7}, E_{6})

the Bruhat ordering when J = ∅ or S = ∅, but is different from the Bruhat ordering in general.

7.2. KLV polynomials. For regular blocks, it is possible to compute Ext-groups, and hence 𝔲-cohomology, using Kazhdan-Lusztig polynomials. It turns out that this is also possible for singular blocks. Fix a singular block O_{S}^{μ} and, as before, let ^{S}W^{J} denote the set that parametrizes the simple modules in O_{S}^{μ}. Then for x, w ∈ ^{S}W^{J} define the Kazhdan-Lusztig-Vogan polynomial ^{S}P_{x,w}^{J} by

SP_{x,w}^{J} = ∑_{i≥0} q^{\frac{l(w)−l(x)−i}{2}} dim Ext_{O_{S}}^{i}(N_{x}, L_{w}).

There is a beautiful formula due to Soergel [Soe] and Irving [Irv] that allows one to compute the ^{S}P_{x,w}^{J} in terms of (parabolic regular) KLV polynomials. It says that for x, w ∈ ^{S}W^{J},

SP_{x,w}^{J} = ∑_{z∈W_{J}}(−1)^{l(z)} ^{S}P_{xz,w} .

Note that if x ∈ ^{S}W^{J} and z ∈ W_{J} then l(xz) = l(x) + l(z). Hence the formula above is equivalent to

dim Ext_{O_{S}}^{i}(N_{w_{S}x·μ}, L_{w_{S}w·μ}) = ∑_{z∈W_{J}}(−1)^{l(z)} dim Ext_{O_{S}}^{i−l(z)}(N_{w_{S}xz·λ}, L_{w_{S}w·λ})

for all i, where λ is any weight such that λ + ρ is regular and anti-dominant integral, e.g., λ = −2ρ. In [Soe], this formula is proved for singular blocks in ordinary category O, i.e., for S = ∅. The general formula follows from the following isomorphism (which is proved using a Lyndon-Hochschild-Serre spectral sequence argument as in [ES1, Chap. 15])

Ext_{O_{S}}^{i}(N_{w_{S}y·μ}, L_{w_{S}w·μ}) ≃ Ext_{O}^{i}(M_{w_{S}y·μ}, L_{w_{S}w·μ}),

where M_{w_{S}y·μ} is the ordinary Verma module of highest weight w_{S}y · μ in the block O^{μ} of ordinary category O.

7.3. A new ordering on ^{S}W^{J}. Using the formula for KLV polynomials ^{S}P_{x,w}^{J} from above, we started to identify weak Kostant modules in singular blocks for simple Lie algebras of small rank. We then noticed that in certain singular blocks, there exist simple modules L_{w} that “ought” to be weak Kostant modules since all non-zero KLV polynomials ^{S}P_{x,w}^{J} = 1, but for which the set {x ∈ ^{S}W^{J} | ^{S}P_{x,w}^{J} = 1} is not an interval of ^{S}W^{J}. This led us to consider a different ordering on ^{S}W^{J}. We propose to use this new ordering to obtain the rank function used to define (weak) Kostant modules (Definition 3.2) for singular blocks of category O_{S}.

Definition. For x < w in ^{S}W^{J} define μ_{S}(x, w) by

μ_{S}(x, w) = dim Ext_{O_{S}}^{1}(L_{x}, L_{w}) = dim Ext_{O_{S}}^{1}(N_{x}, L_{w}).

For x, w ∈ ^{S}W^{J}, write x →_{μ} w if x < w in the Bruhat ordering, μ_{S}(x, w) ≠ 0 and there is no x < z < w in ^{S}W^{J} with μ_{S}(z, w) ≠ 0. Then define ≤_{μ} as the ordering on ^{S}W^{J} generated by the covering relations x →_{μ} w. We call this ordering on ^{S}W^{J} the μ-ordering or the Ext^{1}-ordering.

Remark. Note that by definition, x ≤_{μ} w implies x ≤ w in the Bruhat ordering. If J = ∅ or S = ∅ then the Ext^{1}-ordering and the Bruhat ordering coincide. The proof is easy in the case when J = ∅. The case when S = ∅ follows via Koszul duality for parabolic and singular category O (cf. [Bac] and [BGS]).

We now give two examples to show the usefulness of the new ordering.

7.4. Splitting of singular blocks for Hermitian symmetric cases. In Section 6 (cf. Table 2), we pointed out that for (Φ, Φ_{S}) = (B_{n}, B_{n−1}) or (C_{n}, A_{n−1}), certain singular blocks O_{S}^{μ} split further into a direct sum O_{S}^{μ} = O_{1} ⊕ O_{2} with each of the summands equivalent to a regular block of another Hermitian symmetric pair. The Ext^{1}-ordering reflects this nicely in the sense that the poset^{S}W^{J} splits up into two disjoint posets, each of which looks like a regular parabolic poset. On the other hand, the Bruhat ordering makes ^{S}W^{J} a connected poset. A simple module L_{w} ∈ O_{S}^{μ} which correspond to a Kostant module in O_{1} or O_{2} “ought” to be a Kostant module, but in general the subset of ^{S}W^{J} parametrizing the cohomology of L_{w} will only be an interval in the Ext^{1}-ordering, not in the Bruhat ordering.

7.5. A semi-regular block for (F_{4}, C_{3}). An interesting non-Hermitian symmetric example is obtained for (Φ, Φ_{S}) = (F_{4}, C_{3}). Consider the category O(F_{4}, {b, c, d}, {d}), where the labeling of the simple roots is the same as in Figure 3. The Hasse diagram of the poset ^{S}W^{J} with respect to the usual Bruhat ordering is given on the left of Figure 6. (Note that in the Hasse diagram of ^{S}W, a node that corresponds to an element of ^{S}W^{J} is a node having an edge with label d going up from it.) A calculation shows that Ext^{1}(V_{3}, L_{4}) = 0 and hence the Ext^{1}-ordering on ^{S}W^{J} is different from the Bruhat ordering. The Hasse diagram corresponding to the Ext^{1}-poset is shown on the right side of Figure 6. The structure of

1 2 3 ? 4 5 ? 6  1 2 3  4 5 6

Figure 6. The Bruhat and Ext^{1} posets for (F_{4}, {b, c, d}, {d})

the category O(F_{4}, {b, c, d}, {d}) is exactly the same as O(A_{3}, A_{1} × A_{1}, ∅). Not only are the posets isomorphic, but also the radical filtrations of the generalized Verma modules in the two categories correspond:

N_{1} = L_{1}, N_{2} = \frac{L_{2}}{L_{1}} , N_{3} = \frac{L_{3}}{L_{2}} , N_{4} = \frac{L_{4}}{L_{2}} , N_{5} = \begin{matrix}L_{5} \\ L_{4} L_{3} L_{1} \\ L_{2} \end{matrix} , N_{6} = \begin{matrix}L_{6} \\ L_{5} \\ L_{1} \end{matrix} .

So, H^{0}(𝔲, L_{4}) = F_{4}, H^{1}(𝔲, L_{4}) = F_{2}, H^{2}(𝔲, L_{4}) = F_{1}, and H^{i}(𝔲, L_{4}) = 0 for i > 2. Likewise, H^{0}(𝔲, L_{6}) = F_{6}, H^{1}(𝔲, L_{6}) = F_{5}, H^{2}(𝔲, L_{6}) = F_{3} ⊕ F_{4}, H^{3}(𝔲, L_{6}) = F_{2}, H^{4}(𝔲, L_{6}) = F_{1}, and H^{i}(𝔲, L_{4}) = 0 for i > 4. In particular, using the rank function of the Ext^{1}-poset, L_{4} and L_{6} become Kostant modules.

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Department of Mathematics, University of Georgia, Athens, Georgia 30602

E-mail address: brian@math.uga.edu

Department of Mathematics, Baylor University, Waco, Texas 76798

E-mail address: Markus Hunziker@baylor.edu