Rigidity of generalized Verma modules

Oleksandr Khomenko and Volodymyr Mazorchuk ^{∗}

Abstract

We prove that generalized Verma modules, induced from generic Gelfand-Zetlin modules, and generalized Verma modules, associated with Enright-complete modules, are rigid. Their Loewy length and quotients of the unique Loewy filtrations are calculated for the regular block of the corresponding category O(𝔭, Λ).

1 Introduction

Let 𝔤 be a semi-simple complex finite-dimensional Lie algebra with a fixed triangular decomposition, 𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫_{+}, and 𝔭 ⊃ 𝔥 ⊕ 𝔫_{+} be its parabolic subalgebra. In what follows there will be a distinguished special case, namely when the semi-simple part 𝔞 of the Levi factor 𝔞′ is isomorphic to a direct product of some 𝔰l(n_{i}, ℂ). This will be assume all the time when we will discuss generic Gelfand-Zetlin modules. Let 𝔫 be the nilpotent radical of 𝔭.

This paper continues the study of certain parabolic generalizations, O(𝔭, Λ), of the celebrated BGG category O of 𝔤-modules ([BGG2]). These categories, associated with 𝔭 and certain admissible category Λ of 𝔞-modules, were introduced in [FKM1], where it was shown that their blocks correspond to the so-called standardly stratified algebras in the sense of [CPS] (and even to the smaller class of properly stratified algebras, which was recently introduced in [Dl]). In particular, there is an analogue of the BGG-reciprocity formula, which involves besides the indecomposable projective modules and simple modules a class of intermediate modules, which are called generalized Verma modules. Later in [FKM2, FKM3] the algebra of the principal block of O(𝔭, Λ) was given a combinatorial description, analogous to Soergel’s description of O, [S].

The basic example of O(𝔭, Λ) was constructed in terms of the so-called generic Gelfand-Zetlin modules (see [DOF1]). In [KM1, KM2] this example was related to a certain subcategory of O, which can be described in terms of Enright’s completion functors ([E]). The last categories carry a “strange” abelian structure, which is not inherited from that on O. Where this structure appears from was explained in [KM2] after realizing (Mathieu’s version, [M], of) Enright’s completion functor via the approximation functor with respect to a suitably chosen injective module in O.

∗Junior Associate of the ICTP

From [I2, I3] (combined with the reduction to integral case in [S]) it is known that all Verma modules and big projective modules in the principal blocks of O (this can be viewed as the first extremal case of O(𝔭, Λ), which corresponds to 𝔞 = 0) are rigid, i.e. their socle and radical filtrations coincide and form the unique Loewy filtration (the shortest filtration with semi-simple quotients). In [BGS] this rigidity result was reobtained and extended to all Verma modules using the machinery of Koszul rings and some deep geometrical results. Let O(𝔭, Λ) be the basic example mentioned above. The blocks of O(𝔭, Λ) contain generalized Verma modules induced from generic Gelfand-Zetlin modules and big projective modules. So, there arise a natural question: are these modules also rigid? By [KM1] there is a full functor, say F, from O(𝔭, Λ) to O, which sends each generalized Verma module to a Verma module and each big projective (in O(𝔭, Λ)) to a big projective in O. The main problem is that the abelian structure on the image of F does not coincide with the abelian structure on O. Hence the results of [I2, I3, BGS] can not be applied directly.

The aim of the paper is to give a positive answer to the first part of the question above. Our main result is the following.

Theorem 1. Let Λ be the admissible category, generated by a simple generic Gelfand-Zetlin module, or the admissible category of Enright-complete modules, and O(𝔭, Λ) be the corresponding parabolic analogue of O, [FKM1]. Then generalized Verma modules in O(𝔭, Λ) are rigid.

In contrast with the classical case, the big projective modules in O(𝔭, Λ) fail to be rigid in the general case. We present an sl(3)-counterexample in Section 4. We also remark that, in the second extremal case, namely 𝔤 = 𝔞, all GVMs are simple, hence rigid of Loewy length 1 and the rigidity of the big projective module (=the unique indecomposable projective in the principal block, =the standard module) was proved already in [KM1].

The paper is organized as follows. The next section contains all necessary preliminary information, in particular, we define O(𝔭, Λ) and recall how the situation is related to the category O. Section 3 is devoted to the study of Loewy series on generalized Verma modules. Here we prove rigidity, calculate Loewy length and quotients of the unique Loewy filtration. In Section 4 we compute an 𝔰l(3, ℂ)-example, and, in particular, give an example of a non-rigid big projective module in O(𝔭, Λ). We finish the paper with a short discussion of properties of standard modules in Section 5.

2 Notation and preliminary results

2.1 Generalized Verma modules and generic Gelfand-Zetlin modules

By [KM1], the category constructed here is a special case of the categories considered in the next subsection (we recall that for Gelfand-Zetlin modules we assume that 𝔞 is a product of 𝔰l(n_{i}, ℂ)). The advantage of this construction is that for these categories the notions of simple objects and simple 𝔤-modules coincide.

Let V be an 𝔞′-module. Extending it trivially to a 𝔭-module we can form the module M_{𝔭}(V) = U(𝔤) ⊗_{U(𝔭)} V, which is called a generalized Verma module (GVM in the sequel), provided V is simple.

Let [l] = (l_{i,j})_{i=1,...,n}^{j=1,...,i} be a tableau with complex entries such that l_{i,j} − l_{i,k} ∉ ℤ for all i < n and all possible j, k, and V ([l]) be the corresponding generic Gelfand-Zetlin 𝔤l(n, ℂ)- module (generic GZ-module in the sequel), as defined in [DOF1]. V ([l]) can be restricted in a natural way to the canonical copy of 𝔰l(n, ℂ) ⊂ 𝔤l(n, ℂ) and thus we can talk about generic GZ-modules over 𝔰l(n, ℂ). Now over a direct product of some 𝔰l(n_{i}, ℂ) (e.g. over 𝔞) we define generic GZ-modules as tensor product of the above generic GZ-modules over the components.

Fix a simple generic GZ-module, V, over 𝔞 and consider the category Λ of all subquotients in V ⊗ F, where F runs through all finite-dimensional 𝔤l(n, ℂ)-module. This category extends in a natural diagonalizable way to 𝔞′. The blocks of this category will be module categories over local finite-dimensional associative algebras, [FKM1]. For a simple object, V ∈ Λ, we denote by Ṽ the projective cover of V.

Following [FKM1] we define O(𝔭, Λ) as the full subcategory of the category of all 𝔤modules, which consists of finitely-generated 𝔫-locally finite 𝔤-modules, which decompose into a direct sum of modules from Λ, when viewed as 𝔞′-modules. The blocks of O(𝔭, Λ) are module categories over properly stratified finite-dimensional associative algebras, [FKM1]. GVMs M_{𝔭}(V), V is a simple generic GZ-module over 𝔞′, are objects of O(𝔭, Λ) and they are also proper standard modules for the corresponding properly stratified algebras. The modules Δ(V) = M_{𝔭}(Ṽ), V is simple in Λ, are standard modules for O(𝔭, Λ).

2.2 Enright completions and S-subcategories in O

Let ℜ be the root system of 𝔤 with respect to 𝔥 and π be the basis of ℜ, which corresponds to our fixed triangular decomposition. Fix some Weyl-Chevalley basis, {X_{α}|α ∈ ℜ}, {H_{α}|α ∈ π}, in 𝔤. Denote by W the Weyl group of ℜ and by s_{α} the reflection, corresponding to α ∈ ℜ. We denote by l : W → ℤ_{+} the length function and by ŵ the unique longest element of W.

For a simple root, α, denote by U_{α} the localization of U(𝔤) with respect to {X_{−α}^{k}|k ∈ ℕ} (see [M]) and define the completion functor E_{α} : O → O as the composition of U_{α} ⊗_{U(𝔤)} _{−}, −|_{U(𝔤)} and the functor of taking the locally X_{α}-finite part of a module (see [FKM2, FKM3, KM1, KM2]). Clearly E_{α} ∘ E_{α} = E_{α}.

For S ⊂ π let O(S) denote the full subcategory of O, which consists of all modules on which all X_{−α}, α ∈ S, act injectively. Then for α, β ∈ S the restrictions of E_{α} and E_{β} on O(S) satisfy braid relations (see [De] or [KM2] for a short proof). Hence, if s_{α_{1}} … s_{α_{k}} = w ∈ W_{S} = ⟨s_{α}|α ∈ S⟩ the functor E_{w} = E_{α_{1}} ∘ ⋯ ∘ E_{α_{k}} : O(S) → O(S) is well-defined. Let w_{S} ∈ W_{S} be the longest element. The functor E_{w_{S}} is called the S-completion functor and a module, M ∈ O, is called S-complete if M ∈ E_{w_{S}}(O(S)). The category O(S)_{st} of all S-complete modules is an abelian category with usual kernels and cokernels defined as follows: if f : M → N, M, N ∈ O(S)_{st}, then the cokernel of f is E_{w_{S}}(N/E_{w_{S}}(f(M))) ([FKM3], see [KM2] for another description in terms of approximation with respect to an injective module).

Now let S be the set of simple roots of 𝔞. Then, by [KM1], for appropriate 𝔞, there exists a blockwise equivalence of categories, F : O(𝔭, Λ) → O(S)_{st}, which sends GVMs to Verma modules. Again by [KM1], the functor F^{−1} : O(S)_{st} → O(𝔭, Λ) is also well-defined. In particular, we can set Λ = O(S)_{st} and consider O(𝔭, Λ) in the case of arbitrary 𝔞. We emphasize once more that the advantage of Gelfand-Zetlin modules is that in this case simple objects of O(𝔭, Λ) are simple 𝔤-modules. Finally, we recall the following properties of E_{w_{S}}, [FKM3]:

1. for a simple L ∈ O the module E_{w_{S}} is either 0 or simple in O(S)_{st};

2. E_{w_{S}}(M) ⊂ E_{w_{S}}(N) for M ⊂ N;

3. E_{w_{S}}(N/M) ⊃ E_{w_{S}}(N)/E_{w_{S}}(M) for M ⊂ N.

2.3 Loewy length and Loewy filtrations

Let M be a module of finite length. A filtration, 0 ⊂ M_{1} ⊂ ⋯ ⊂ M_{k} = M is called a Loewy filtration if all M_{i}/M_{i−1} are semi-simple and k is the minimal possible. This k is called the Loewy length of M and is denoted by ll(M) (if M is an object of two different categories and we want to underline that it’s Loewy length is considered inside, say the category A, we will write ll_{A}(M)). Associated with M there are two Loewy filtrations: the socle filtration 0 ⊂ soc^{1} M ⊂ soc^{2} M ⊂ ⋯ ⊂ soc^{ll(M)} = M (here soc^{i+1} M/soc^{i}M = soc(M/ soc^{i} M)) and the radical filtration 0 = rad^{ll(M)} M ⊂ ⋯ ⊂ rad^{1} M ⊂ rad^{0} M = M (here rad^{i}(M) = rad(rad^{i−1} M)). If 0 ⊂ M_{1} ⊂ ⋯ ⊂ M_{k} = M is a Loewy filtration of M, there holds rad^{ll(M)−i} M ⊂ M_{i} ⊂ soc^{i} M. The module M is called rigid if the socle filtration and the radical filtration coincide and thus there is only one Loewy filtration. We will denote the layers of the socle and the radical filtrations by soc_{i} M = soc^{i} M/ soc^{i−1} M and rad_{i} M = rad^{i} M/ rad^{i+1} M correspondingly.

2.4 Soergel’s combinatorics of O and its generalization to O(𝔭, Λ)

For λ ∈ 𝔥^{∗} we will denote by M(λ) the Verma module with highest weight λ, by L(λ) the unique simple quotient of M(λ), by P (λ) the indecomposable projective cover of L(λ) and by I(λ) the indecomposable injective envelope of L(λ). If ρ is the half-sum of all positive root, we recall the standard ·-action of W on 𝔥^{∗}, defined as follows: w · (λ) = w(λ + ρ) − ρ.

Let χ be an integral central character of 𝔤 and O_{χ} be the block of O, corresponding to χ. By the main result of [S], any indecomposable block of O is equivalent to O_{χ} for some χ, however, this equivalence may force to change 𝔤. The character χ thus corresponds to an integral W -orbit, W · λ, where λ is antidominant. Let W_{λ} denote the stabilizer of λ. Then the simples in O_{χ} are naturally parameterized by cosets ξ ∈ W/W_{λ}. If w ∈ ξ is the longest element, then we set L(ξ) = L(w · λ). Analogously we define P (ξ) = P (w · λ), Δ(ξ) = Δ(w · λ) and M(ξ) = M(w · λ). Set P_{λ} = ⊕_{ξ∈W/W_{λ}}P (ξ) and A_{λ} = End_{O}(P_{λ}). A_{λ} is a basic, finite-dimensional, associative and quasi-hereditary algebra. The Verma modules M(ξ) are standard modules with respect to the quasi-hereditary structure. Set C_{λ} = End_{O}(P (λ)). By [S, Endomorphismensatz 7], the algebra C_{λ} is the subalgebra of W_{λ}-invariants in the coinvariant algebra C, which is the quotient of ℂ[𝔥^{∗}] modulo the ideal generated by W -invariant, with respect to the usual W -action, polynomials without constant term. In particular, C_{λ} = C if λ is regular. Let e be the primitive idempotent of A_{λ}, corresponding to ξ = W_{λ} (which means C_{λ} = End_{A_{λ}Ae}). Then, by [S, Struktursatz 9], A_{λ} ≃ End(A_{λ}e_{C_{λ}}). This is usually called the double centralizer property. For a primitive idempotent, f, of A_{λ} we will denote the C_{λ}-module fAe by D_{f}.

Now we recall the analogous description of O(𝔭, Λ), the latter as defined in Subsection 2.2 ([FKM2, FKM3]). For this we consider the equivalent category O(S)_{st} and set O_{λ}^{S} = O(S)_{st} ∩ O_{λ}. We emphasize that the abelian structure on O_{λ}^{S} is not inherited from that on O_{λ}. The simples of O_{λ}^{S} are indexed by those ξ ∈ W/W_{λ} such that the longest representative w of ξ is at the same time the shortest element in a coset from W/W_{S}. Let T = {ξ_{1}, …, ξ_{k}} be the complete set of parameters of simples in O_{λ}^{S}. Set P_{λ}^{S} = ⊕_{ξ∈T} P (ξ) and A_{λ}^{S} = End_{𝔤}(P_{λ}^{S}). Denote by e^{S} the primitive idempotent of A_{λ}^{S} corresponding to P (λ). Then, by [FKM3, Section 5], we have A_{λ}^{S} = End((A_{λ}^{S}e^{S})_{C_{λ}}).

3 Rigidity of generalized Verma modules

In this section we prove the main result of this paper, the rigidity of GVMs. Our proof follows the ideas of [BGS]. In fact, we are going to realize GVMs as graded modules over graded algebras and then apply the following lemma ([BGS, Proposition 2.4.1]).

Lemma 1. Let R be a ℤ-graded ring, generated by R_{0} and R_{1}, such that R_{0} is semi simple. Let M be a graded R-module of finite length. If soc(M) (resp. rad(M)) is simple, then the socle (resp. radical) filtration on M coincides with the grading filtration, up to shift.

The coinvariant algebra C is graded in a natural way, and we fix a grading, in which the generators (𝔥^{∗}) have degree 2. Hence the algebra C_{λ} is graded as well. Both modules A_{λ}e and A_{λ}^{S}e^{S} are graded C_{λ}-modules (see Remark after [S, Lemma 7]). Moreover, as A_{λ}^{S} = tA_{λ}t for some, in general case non-primitive, idempotent t of A_{λ}, we can naturally consider A_{λ}^{S}e^{S} as a graded submodule of A_{λ}e. The decomposition A_{λ}e = ⊕_{f}D_{f}, where f runs through the set of all primitive idempotents of A_{λ}, is a decomposition of A_{λ}e into a direct sum of indecomposable modules and all summands are graded. If f corresponds to L(ξ), ξ ∈ W/W_{λ}, then the minimal non-zero degree of D_{f} is exactly ll(M(ξ)). Via the double centralizer property we get a positive grading on A_{λ}, which corresponds to the “mixed” structure on O. By [BGS, Section 4], this grading is Koszul, in particular, (A_{λ})_{0} is semi simple and A_{λ} is generated by (A_{λ})_{1} over (A_{λ})_{0}.

Now we want to move this picture to A_{λ}^{S}, which is a positively graded subalgebra of A_{λ} via the double centralizer property. This trivially implies that (A_{λ}^{S})_{0} is semi simple as well. Unfortunately, in the general case A_{λ}^{S} is not generated by (A_{λ}^{S})_{1} over (A_{λ}^{S})_{0}. We refer the reader to Section 4 for the corresponding 𝔰l(3, ℂ)-example. However, we can find some convenient set of generators for A_{λ}^{S}.

Since the algebra C_{λ} is commutative the double centralizer property implies that it is a subalgebra of A_{λ}^{S} and, moreover, is a subalgebra of the center of A_{λ}^{S}. We also recall that C_{λ} appears in [S] as the image of the action of Z(𝔤) on P (λ).

Proposition 1. A_{λ}^{S} is generated by (A_{λ}^{S})_{0}, (A_{λ}^{S})_{1} and C_{λ}.

Proof. Let f_{1} and f_{2} be two primitive idempotents of A_{λ}^{S}. Our aim is to decompose any graded element from f_{1}A_{λ}^{S}f_{2} into a product of elements from (A_{λ}^{S})_{0}, (A_{λ}^{S})_{1} and C_{λ}. But any element from f_{1}A_{λ}^{S}f_{2} corresponds to a map from an indecomposable projective P (ξ_{1}) to indecomposable projective P (ξ_{2}) in O(𝔭, Λ). from the definition of projective modules it follows that the map φ : P (ξ_{1}) → P (ξ_{2}) can not be decomposed non-trivially into a product of other maps between projectives only in the case, when φ(P (ξ_{1})) ∉ rad^{2}(P (ξ_{2})). Hence, it is enough to prove that, under this assumption, we have φ ∈ (A_{λ}^{S})_{0} or φ ∈ (A_{λ}^{S})_{1} or φ ∈ C_{λ}. We have to consider several cases.

First, let ξ_{1} = ξ_{2} and φ be an isomorphism. Then the assumption that φ is a graded element implies φ ∈ (A_{λ}^{S})_{0}.

Now, let ξ_{1} ≠ ξ_{2}. Because of the duality on O(𝔭, Λ) we may assume ξ_{1} ≰ ξ_{2} with respect to the order of properly stratified structure. Then, necessarily L(ξ_{1}) is a top of rad(P (ξ_{2})). Now we recall that, according to [FKM1], all projectives in O(𝔭, Λ) are filtered by standard modules. Because of ξ_{1} ≰ ξ_{2} and the BGG-reciprocity, [FKM1], the standard module Δ(ξ_{1}) occurs in this filtration. Hence there should exist a quotient of P (ξ_{2}), which is an extension of Δ(ξ_{1}) by Δ(ξ_{2}), such that the unique copy of L(ξ_{1}) in the top of Δ(ξ_{1}) is exactly the one, covered by φ. But this means that φ covers the whole Δ(ξ_{1}) as it has a simple top. So, the image of φ does not belong to rad(P (ξ_{2})) in the category O as well. This implies that φ is a degree 1 map in A_{λ}, as it is graded. Hence φ ∈ (A_{λ}^{S})_{1}.

Finally, we consider the case ξ_{1} = ξ_{2} = ξ and φ(P (ξ)) ⊂ rad(P (ξ)). Again we recall that P (ξ) is filtered by standard modules and there is only one of them (the last one), isomorphic to Δ(ξ). As all other standard modules have different tops, φ maps the top of P (ξ), which is also the top of Δ(ξ), into some other copy of L(ξ) in Δ(ξ). Our claim will easily follow if we prove that the natural action of C_{λ} on Δ(ξ) surjects onto End_{A_{λ}}(Δ(ξ)). Realize Δ(ξ) as M_{𝔭}(Ṽ), where Ṽ is projective in Λ. Then the exactness of induction guarantees End_{A_{λ}}(Δ(ξ)) = End_{Λ}(Ṽ). The S-Harish-Chandra homomorphism, [DOF2], restricts the action of Z(𝔤) on Δ(ξ) to the action of Z(𝔞) on Ṽ, and the statement follows from [S, Endomorphismensatz]. This completes the proof.

Lemma 2. (C_{λ})_{>0} annihilates M(ξ).

Proof. The action of C_{λ} comes from the action of Z(𝔤), which acts by scalars on M(ξ), [FC, DOF2].

Lemma 3. M(ξ) is a graded A_{λ}^{S}-module.

Proof. It follows directly from [BGS], as M(ξ) is a graded A_{λ}-module and A_{λ}^{S} is a graded subalgebra of A_{λ}. But one can also get this from our graded picture and the doublecentralizer. Obviously, A_{λ}^{S}e^{S} is a positively graded finite-dimensional A_{λ}^{S}-module. Moreover, it is filtered by GVMs and, by BGG-reciprocity, [FKM1], each GVM from O_{λ}^{S} occurs as a subquotient of A_{λ}^{S}e^{S}. The biggest GVM is naturally identified with soc(A_{λ}^{S}e^{S}), the latter viewed as C_{λ}-module, hence graded. The statement follows by induction.

We are now ready to state our main result.

Theorem 2. Any GVM M(ξ) in O_{λ}^{S} is rigid.

Proof. By Lemma 3, M(ξ) is a graded module over graded algebra A_{λ}^{S}. By Lemma 2, it is even a graded module over a graded algebra A_{λ}^{S}/((C_{λ})_{>0}). By Proposition 1, the last is generated in degrees 0 and 1. It is well-known that GVMs have simple socles and tops, [FC, MO]. Hence the statement follows from Lemma 1.

Now we can calculate the Loewy length of GVMs for regular λ. We recall that in this case they are parameterized by ξ ∈ W/W_{S}. We start with the following easy observation.

Lemma 4. The inequality ll_{O(𝔭,Λ)}(F^{−1} ∘ E_{w_{S}}(M)) ≤ ll_{O}(M) is true for any M ∈ O.

Proof. Follows from properties of E_{w_{S}}, described in Subsection 2.2.

Now we can calculate the Loewy length of M(ξ).

Lemma 5. Let λ be regular and ξ ∈ W/W_{S}. Then ll(M(ξ)) = l(w^{ξ}) + 1.

Proof. The inequality ll(M(ξ)) ≥ l(w^{ξ}) + 1 follows from the analogue of BGG-criterion for inclusion of GVMs, [MO]. By this criterion, each M(ξ) has a filtration, 0 ⊂ M(ξ_{1}) ⊂ ⋯ ⊂ M(ξ_{r}) = M(ξ), where r = l(w^{ξ}) + 1 and all quotients of the filtration are non-zero. As each GVM has a simple top, M(ξ_{i}) lies in the radical of M(ξ_{i+1}) and hence ll(M(ξ)) ≥ r.

Let us now prove that ll(M(ξ)) ≤ l(w^{ξ}) + 1. Consider two Verma modules M(w_{ξ}λ) and M(w^{ξ}λ). By [De] E_{w_{S}}(M(w^{ξ}λ)) = M(w_{ξ}λ) and, as we have already mentioned, F^{−1}(M(w_{ξ}λ)) = M(ξ). Hence, by Lemma 4, we have ll_{O(𝔭,Λ)}(M(ξ)) ≤ ll_{O}(M(w_{ξ}λ)) = l(w^{ξ}) + 1. This completes the proof for GVMs.

We remark that in the case of singular λ one can also calculate the Loewy length in terms of height of an ideal in the poset W/W_{λ}. However, this does not give any closed formula as in Lemma 5.

Combining the above results we can describe the layers of the unique Loewy filtration of M(ξ).

Corollary 1. Let λ be regular, ξ ∈ W/W_{S}, w the shortest representative of ξ, and i ∈ ℤ_{+}. Then soc_{i}(M(ξ)) = rad_{ll(M(ξ))−i+1}(M(ξ)) = F^{−1} ∘ E_{w_{S}}(soc_{i}(M(w · λ))).

Proof. As it was already mentioned, F^{−1} ∘E_{w_{S}}, sends any Loewy filtration of M(w ·λ) to a filtration of M(ξ) with semi-simple subquotients. But by Lemma 5, we have ll_{O}(M(w·λ)) = ll_{O(𝔭,Λ)}(M(ξ)) and both modules are rigid. The statement follows.

4 𝔤 = 𝔰l(3, ℂ), 𝔞 = 𝔰l(2, ℂ) – example

In this section we assume 𝔤 = 𝔰l(3, ℂ) with the standard Cartan subalgebra and {α, β} is a basis of Δ. We assume that 𝔞 = 𝔰l(2, ℂ) is the subalgebra corresponding to the root α. We fix λ regular antidominant. Then O_{λ} contains 6 simple modules indexed by 1 = λ, 2 = s_{α} · λ, 3 = s_{β} · λ, 4 = s_{α}s_{β} · λ, 5 = s_{β}s_{α} · λ and 6 = s_{α}s_{β}s_{α} · λ. The Verma modules have the following radical filtrations (the number on the left is the index i of rad_{i}):

M(1)  M(2)  M(3)  M(4)  M(5)  M(6)
0  1  2  3  4  5  6
1  1  1  2 3  2 3  4 5
2  1  1  2 3
3  1

The big projective module A_{λ}e is rigid and filtered by all Vermas with all multiplicities equal to 1. The graded picture of A_{λ}e as C_{λ} module is the following one, where by i we denote the simple subquotients, which come from 𝔤-simple subquotients of A_{λ}, isomorphic to L(i). This is what we called D_{f} before. Here we will call these C_{λ}-direct summands of A_{λ}e by D_{i} respectively. The number of the left is the degree of the corresponding graded component.

0  1
1  2  3
2  1 1  4  5
3  2 2  3 3  6
4  1 1  4  5
5  2  3
6  1

In particular, the algebra C_{λ}, as a graded left module over itself, is isomorphic to D_{1}. Going to O(𝔭, Λ) we take (see [FKM3, KM2]) the category of injectively copresented modules with respect to the direct sum of indecomposable injectives indexed by the shortest representatives in W/W_{S}. This means that the only indices, corresponding to these shortest representatives, namely 1, 3 and 5, will survive. Hence, the graded structure of A_{λ}^{S}e^{S} will be:

0  1
1  3
2 1 1 5
3  3 3
4 1 1 5
5  3
6  1

Now let us look at GVMs and standard modules, which we will naturally index by 1, 3 and 5. As all multiplicities are one, all GVMs are uniserial and have the following radical filtrations:

M_{𝔭}(1)  M_{𝔭}(3)  M_{𝔭}(5)
0  1  3  5
1  1  3
2  1

To obtain Δ(5) one has to find the trace of D_{5} in A_{λ}^{S}e^{S}. It is easy to see that this will be the following part of A_{λ}^{S}e^{S}:

2  5
3  3
4 1 5
5  3
6  1

Analogously, Δ(3) is presented by the following part of A_{λ}^{S}e^{S}:

1  3
2  1
3  3
4  1

By direct calculation one easily gets that these graded filtrations are in fact Loewy filtrations of the corresponding standard modules. The module Δ(1) is, obviously, a selfextension of M_{𝔭}(1). It is presented by the following part of A_{λ}^{S}e^{S}:

0  1 1 2  1

And here we see the difference: to obtain the radical filtration of Δ(1) one has to re-scale the grading. The last causes the fact that the big projective module A_{λ}^{S}e^{S} in O(𝔭, Λ) is not rigid. Indeed, let us look at the first three graded components of it:

0  1
1  3
2  1 1

Clearly, the dimension of C_{λ}-homomorphisms of degree 1 from D_{1} to D_{3} is 1. But the dimension of C_{λ}-homomorphisms of degree 1 from D_{3} to D_{1} is also 1 (e.g since D_{1} is injective). Hence their composition has dimension at most 1, which means that one composition factor of rad_{1}(D_{1}) belongs to rad_{1}(A_{λ}^{S}e^{S}). The above description of Loewy filtrations of standard modules (it is enough to have it for Δ(5)) implies that the Loewy length of A_{λ}^{S}e^{S} is 7. hence the graded filtration is a Loewy one and does not coincide with the radical filtration. Therefore A_{λ}^{S}e^{S} is not rigid.

5 What can be said about standard modules?

Based on the example above we formulate the following conjecture: Conjecture. Standard modules in O(𝔭, Λ) are rigid.

We also have to remark that even the calculation of Loewy length of standard modules seems to be a very non-trivial problem. As the above example shows, the answer will be much more difficult than that for GVMs. However, it is not difficult to get the following inequalities:

Lemma 6. Let M = M_{𝔭}(V) be a generalized Verma module and Δ = Δ(Ṽ) be the corresponding standard module. Then

ll(M) + ll(Ṽ) − 1 ≤ ll(Δ) ≤ ll(M) + 2ll(Ṽ) − 1.

Proof. To prove the left inequality we recall that Ṽ has simple socle, V, and M is a submodule of Δ with simple top L_{𝔭}(V) and all other composition factors different from L_{𝔭}(V). Hence M ⊂ rad_{ll(Ṽ)−1}(Δ), which implies necessary inequality.

The right inequality follows from the double centralizer property. Indeed, it implies that the big projective is graded as A_{λ}-module. Moreover, by BGG reciprocity, it is filtered by standard modules, and by induction one derives that all standard modules are in fact graded modules over A_{λ}. It is straightforward to see that the length of the grading filtration inherited from the big projective module is precisely ll(Δ) ≤ ll(M) + 2ll(Ṽ) − 1. Indeed, ll(Δ) ≤ ll(M) corresponds to the Verma submodule M of Δ and its grading filtration and 2ll(Ṽ) corresponds to the grading filtration of Ṽ as C_{λ}-module (we recall that C_{λ} is even-graded). This implies the necessary inequality.

We note that both extremal cases of equalities are possible. This can be read off from the example in Section 4. Indeed, the left equality holds for Δ(1) and the right equality holds for both Δ(3) and Δ(5). It is easy to construct other examples where ll(Δ) satisfies two strict inequalities from the above formula. However, for this one has to take 𝔞 of rank greater than 1.

Acknowledgments

The research was done during the visit of the second author to Max-Planck-Institut für Mathematik in Bonn, Germany. The financial support, hospitality, accommodation and perfect working conditions provided by the Institut are gratefully acknowledged. We also thank Prof. Steffen König for helpful discussions.

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O.K.: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstrasse 1, D-79104, Freiburg im Breisgau, Germany,

e-mail: khomenko@mathematik.uni-freiburg.de

V.M.: Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-5311, Bonn, Germany,

e-mail: mazor@mpim-bonn.mpg.de;

on leave from: Chalmers Tekniska Högskola och Göteborgs Universitet, Göterorg, Swe-

den, e-mail: volody@math.chalmers.se