Translated simple modules for Lie algebras and simple supermodules for Lie superalgebras

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Introduction
For a finite dimensional Lie algebra k, we consider modules of the form S ⊗ E, where S is a simple (but not necessarily finite dimensional) k-module and E is a finite dimensional k-module. The module S ⊗ E is always noetherian and it is natural to ask, see e.g. [32,Section 1.3], whether S ⊗ E is artinian. The latter is true, for example, if k ∼ = sl 2 (C). However, in the general case the answer is negative, see [42,Theorem 4.1] for k = sl 2 (C) ⊕ sl 2 (C).
In the first half of the present paper, we investigate the following two questions.
Q1: Does the quotient of S ⊗ E by its radical have finite length? Q2: Is the socle of S ⊗ E is an essential submodule?
Note that, since the module S ⊗ E is noetherian, its socle must have finite length and its radical must be superfluous. We show that question Q1 has an affirmative answer for arbitrary Lie algebras over arbitrary fields. We also show that Q2 has an affirmative answer for reductive Lie superalgebras over C of type A. For the latter we apply the theory of projective functors of [3] and some specifics about Kazhdan-Lusztig combinatorics in type A. Our interest in Q2 stems from an application to representation theory of Lie superalgebras, which occupies the second half of the paper. Let g be a finite dimensional Lie superalgebra and g0 be the Lie algebra forming the even part of g. A basic problem in the representation theory of g is classification of simple g-supermodules. This problem is, most probably, too difficult in the general case. However, a natural variant of this problem is reduction to classification of simple g0-modules. In [10] it was shown that for type I Lie superalgebras there is a natural bijection between simple g-supermodules and simple g0-supermodules (i.e. pairs consisting of a simple g0-module and an element in {0, 1}). Such a nice result seems unrealistic outside type I. However, in the present paper we provide a weaker result (see Theorem 47) for all finite dimensional classical complex Lie superalgebras for which g0 has type A. The crucial ingredient in the proof is the fact that any g-supermodule is a quotient of an induced g0module. And this latter fact follows from the fact that, in case g0 has type A, question Q2 has an affirmative answer (see Proposition 22 in the first half of the paper).
The paper is organized as follows: Sect. 2 collects all necessary preliminaries for the first half of the paper. Section 3 studies socles and radicals in the biggest possible generality. Section 4 concentrates on similar questions for semisimple Lie algebras over C. Section 5 deals with the very specific case of a sum of copies of sl 2 . In Sect. 6 we collected necessary preliminaries about Lie superalgebras. Section 7 is devoted to classification of simple supermodules for Lie superalgebras with type A even part and to the study of rough structure of such supermodules. Finally, Sect. 8 describes behavior of Kac induction functor, with respect to socles and radicals, for Lie superalgebras of type I.

Lie algebras
Let k be a field and k a finite dimensional Lie algebra over k. The universal enveloping algebra of k will be denoted by U := U (k) and the center of U by Z = Z (k). Denote by the set of central characters χ : Z (k) → C. We set m χ = ker χ, for all χ ∈ .
We denote by U -mod the abelian category of all finitely generated left U -modules. For χ ∈ , denote by U -mod χ the full subcategory of U -mod consisting of all modules on which the action of m χ is locally nilpotent. Set For each χ ∈ , we denote by Pr χ the projection from U -mod Z to U -mod χ with respect to the decomposition above.
Let F denote the category of all finite dimensional U -modules. We denote the duality Hom k (−, k) on F by * , where the left module structure is obtained using the antiautomorphism of U given by X → −X , for X ∈ k.
For a finite dimensional module E, we have the corresponding exact endofunctor F E = −⊗ E of U -mod. The functor F E is left and right adjoint to F E * , see e.g. [3, § 2.1(d)]. In case of a semisimple algebra g, the direct summands of the restriction to U -mod Z of the functors F E are known as projective functors, see [3].
For any M ∈ U -mod, we denote by Ann(M) the two-sided ideal in U that consists of all elements which annihilate every vector in M. The arguments in [27,Kapitel 5] show that, for any M 1 , M 2 ∈ U -mod, E ∈ F and χ ∈ , we have (2.2)

Gelfand-Kirillov dimension and Bernstein number
We fix an arbitrary field k and a finite dimensional Lie algebra k over k. Consider the filtration where U n is spanned, as a vector space, by all products of n or fewer elements of k. By the PBW theorem, the associated graded algebra gr(U ) is isomorphic to the symmetric algebra S(k). This allows us to use an alternative definition of Gelfand-Kirillov dimension, see [33,Section 7], which we recall below. For a (non-zero) finitely generated left U -module M, with generating subspace M 0 , set M n = U n M 0 , for all n ∈ N. There exists n 0 ∈ N, such that we have d ∈ N and {a i ∈ Z, 0 ≤ i ≤ d}, with a d = 0, for which The right-hand side is the Hilbert-Samuel polynomial in n. The degree of this polynomial is the Gelfand-Kirillov dimension GK(M) := d ∈ N and the leading coefficient is the Bernstein number e(M) := a d ∈ Z >0 . These two numbers do not depend on the choice of M 0 . By [27,Lemma 8.8], for any M ∈ U -mod and E ∈ F , we have 3) The following statement can be found in [33,Theorem 7.7]. .
By the discussion at the end of Sect. 2.1, we have the following observation.

Lemma 10
Assume that M ∈ U -mod admits a GK-complete filtration. If N is a subquotient of M with the same Gelfand-Kirillov dimension, then N admits a GK-complete filtration as well.

Triangular decomposition
In this section we assume that g is a semisimple Lie algebra over C. In order to recall the classification of projective functors for g from [3], it is convenient to choose a triangular decomposition of g with h a Cartan subalgebra and b = h ⊕ n + a Borel subalgebra. Denote by ⊂ h * the set of roots of g with respect to h. We have = + − , with + = − − being the roots of n + . For each root α ∈ , we have the coroot h α ∈ h.
Consider the Weyl group W = W (g : h) with its defining action on h * . The group W is generated by s α , where α ∈ , and the action of these generators on h * is given by We have the set of integral weights For each E ∈ F , we denote by supp(E) ⊂ the support of E, that is the set of all h-weights of E. Then we have For a coset ∈ h * / , we have the integral Weyl group W ⊂ W , which is generated by all reflections corresponding to α ∈ for which λ(h α ) ∈ Z, where λ ∈ .
We have the Harish-Chandra isomorphism η * : Z (g) ∼ → S(h) W and an epimorphism The fibers of η are precisely the Weyl group orbits in h * . We call a central character χ ∈ regular if the set η −1 (χ) has size |W |. We call χ integral if η −1 (χ) ⊂ . We introduce the partial order ≤ on h * which is generated by Associated to the triangular decomposition above, we have the BGG category O defined as the full subcategory of U -mod consisting of all weight modules which are locally U (n + )finite, see [4,24]. We denote by ρ the half of the sum of all elements of + . For λ ∈ h * , we have the Verma module λ = U ⊗ U (b) C λ−ρ , induced from the one-dimensional module C λ−ρ of b on which h acts through λ − ρ. The unique simple quotient of λ is denoted by L λ and the projective cover of L λ in O is denoted by P λ .

Projective functors for semisimple Lie algebras
We keep the notation and assumptions of the previous subsection. Following [3,Section 1.4], we have the set We set = 0 /W , for the diagonal action of W on h * × h * . Each class in contains at least one (μ, λ), where λ is dominant and μ is such that μ ≤ wμ, for all w ∈ W λ . We call such a pair a proper representative. The following claim can be found in [3,Theorem 3.3].

Lemma 11
Each projective functor on U -mod Z decomposes into a finite direct sum of indecomposable projective functors. We have a bijection ξ → F(ξ ) between and the set of isomorphism classes of indecomposable projective functors as follows. For each proper representative (μ, λ) of ξ ∈ , the functor F Note that the above lemma implies that the decomposition into indecomposable projective functors is, in fact, unique, up to isomorphism. In case we work with non-integral central characters, we have several ways of denoting the same indecomposable projective functor. The first claim of the following lemma can be found in [3,Theorem 4.1] while the second claim can be found in [27, § 4.13].
Denote by U -mod 0 χ the full subcategory of U -mod χ consisting of all modules M with m χ ⊂ Ann(M). The following claim can be found in [3,Theorem 3.5].

Kazhdan-Lusztig combinatorics and type A
We keep the notation and assumptions of the previous subsection. For a fixed regular dominant λ ∈ , we use the notation θ x := F xλ λ , for all x ∈ W . It then follows from the validity of the Kazhdan-Lusztig conjecture, see [1,8] and Lemma 11, that the composition of the projective functors θ x is governed by Kazhdan Consequently, the Kazhdan-Lusztig preorders of [29, § 1] can be realised by projective functors on a regular block. For convenience, we extend this to the entire category of projective functors. We thus introduce the following preorder on the set of (isomorphism classes of) indecomposable projective functors. We say that F G if G appears as a direct summand of F • F • F , for some projective functors F , F . We have the corresponding equivalence relation, denoted F ∼ G, which means that F G F. Equivalence classes for ∼ are called two-sided cells. Similarly one defines the left and the right preorders and the left and the right cells.
If the Weyl group W is the symmetric group, the preorder can be described in terms of the dominance order on partitions using the Robinson-Schensted correspondence, see [20,Theorem 5.1]. As a consequence, each two-sided Kazhdan-Lusztig cell contains the longest element of some parabolic subgroup. Another consequence (see [20,Corollary 5.6]) is that a left and a right cell inside the same two-sided cell intersect in at most one element. By the above and [20,Theorem 5.3] this translates into two well-known facts for Lie algebras of type A.
Let λ ∈ h * be integral, regular and dominant, and θ an indecomposable projective endofunctor of U -mod η(λ) . Fact 1: There exists x ∈ W , which is the longest element of a parabolic subgroup such that θ ∼ θ x . Fact 2: The only indecomposable projective endofunctor θ of U -mod η(λ) which satisfies θ ∼ θ and appears both as a direct summand in θ • G 1 and G 2 • θ , for some projective functors G 1 , G 2 , is θ itself.
We translate these facts into the formulation that we will require.

Lemma 14
Assume g is of type A and fix a ∼-equivalence class of projective functors. Proof We start from an arbitrary indecomposable projective functor and consider its equivalence class. It follows from Lemma 12(ii) that F ∼ G, for some endofunctor G of U -mod η(λ) , where λ is any fixed dominant regular weight in η −1 (χ 1 ) + . Assume, for simplicity, that λ is integral. The non-integral case is proved using the same arguments, since the integral Weyl group in type A is always of type A. By Fact 1, we have G ∼ θ x 0 , for the longest element x 0 of some parabolic subgroup of W . Take some dominant μ ∈ such that W μ is this parabolic subgroup. Then we have , which concludes the proof of part (i).
To prove part (ii), we assume we have Id η(μ) ∼ H , for some indecomposable projective endofunctor H of U -mod η(μ) . We set The second equation shows that there exists an indecomposable summand H 1 of H 1 for which we have H 1 ∼ H ∼ θ x 0 . By Fact 2, the first equation therefore implies that Applying the second equation again together with Lemma 12(ii) then shows that Id η(μ) must appear as a direct summand in H ⊕|W μ | 2 . This is only possible if H = Id η(μ) . This concludes the proof of part (ii).

Arbitrary Lie algebras
Fix a finite dimensional Lie algebra k over a field k and set U = U (k).

Connection with GK dimension
Theorem 16 Fix a simple U -module S and E ∈ F . Then T := S ⊗ E has finite type radical, and soc(T ) has finite length. Furthermore, the following are equivalent:

c) Every non-zero submodule of T contains a simple subquotient of Gelfand-Kirillov dimension GK(S).
We start the proof with the following lemma.

Lemma 17 With T as in Theorem 16, for any short exact sequence of U -modules
Proof We prove that GK(M 1 ) = GK(S). The statement for M 2 is proved similarly. By Lemma 8(i) and Eq. (2.3), we have GK(M 1 ) ≤ GK(S). By adjunction Hence, S is a quotient of M 1 ⊗E * . Lemma 8(i) and Eq. (2.3) thus imply that GK(S) ≤ GK(M 1 ), which concludes the proof.

Lemma 18
There exists k ∈ Z ≥0 , such that T cannot have a semisimple submodule or quotient of length greater than k.
Proof We prove the claim for quotients, the case of submodules is proved similarly or follows from Remark 2(iii). By Lemma 17, a semisimple quotient of T is a direct sum of simple modules with the GK dimension of each simple equal to GK(S). By Lemma 8(ii), we can thus choose k = e(T ).

Corollary 19
The module T has finite type radical.
Proof Since T is finitely generated, it suffices by Remark 5(ii) to show that the module T /rad(T ) has finite length. If it would have infinite length, we could take an arbitrarily large, but finite, direct sum of simple modules as a quotient of T , which is contradicted by Lemma 18.

Proof of Theorem 16
This first claim is Corollary 19 and Lemma 18. Now we prove the equivalence of (a), (b) and (c). First we prove that (a) implies (b). Take N with non-zero N ⊗ E * → S and consider the corresponding non-zero morphism α : N → S ⊗ E. The image im(α) is a submodule of S ⊗ E and thus has a non-zero socle by assumption. We take a simple module L in that socle, which, by construction, is a subquotient of N . The inclusion L → S ⊗ E yields a non-zero morphism L ⊗ E * → S.
Next we note that (b) trivially implies (c). So, it remains to show that (c) implies (a). We assume (c) holds and set d := GK(S). To obtain a contradiction via Remark 2(ii), we assume we have a non-zero submodule M of T with zero socle. Assume we have [M : L] = 0, for some simple U -module L with GK(L) = d. We thus must have submodules Since M has zero socle, we have M 1 = 0. By construction, M 1 is again a submodule of T with zero socle. By Lemma 17, we have GK(M 1 ) = d = GK(M). By Lemma 8(ii), we have e(M) > e(M 1 ) > 0. This means that, after repeating the above construction M → M 1 a finite number times, we obtain a non-zero submodule N of M (and hence of T ) with zero socle, and such that [N : L] = 0, for all simple U -modules with GK(L) = d. It thus follows that any simple subquotient L of N has Gelfand-Kirillov dimension less than d. Hence assumption (c) yields a contradiction. This completes the proof.

Corollary 20 If T = S ⊗ E admits a GK-complete filtration, it has finite type socle.
Proof By Lemmata 10 and 17, any submodule of T has a GK-complete filtration. Assume that T does not have finite type socle. By Theorem 16, this means that T has a submodule N with GK(N ) = d and such that [N : L] = 0, for all simple modules L with GK(L) = d. Hence N cannot have a GK-complete filtration, a contradiction.

Restriction to blocks
Theorem 21 Let S be a simple U -module with central character χ ∈ . If F(S) has finite type socle, for each projective endofunctor F of U -mod χ , then we have the following: Proof We start by proving Claim (i). We set d = GK(S) and use the equivalence between (a) and (c) in Theorem 16. We thus take an arbitrary submodule N of S ⊗ E. By adjunction, we have a non-zero morphism N ⊗ E * → S, which implies Pr By assumption, the socle of the right-hand side is an essential submodule. Hence we can take a simple submodule L in the socle of the right hand side which is also contained in the left-hand side. Note that GK(L) = d by Lemma 17. So L is a submodule of N ⊗ E * , which leads through adjunction to a non-zero morphism L ⊗ E → N . Since L ⊗ E has finite type radical, there is a simple submodule L 1 of top(L ⊗ E) such that [N : L 1 ] = 0. By Lemma 17, we have GK(L 1 ) = d. Claim (i) thus follows from Theorem 16. Now we consider the set-up of claim (ii). By construction, S ⊗ E is a submodule of S ⊗ V ⊗ E. By claim (i), the latter has finite type socle, completing the proof.

Application to Lie superalgebras
Let s = s0 ⊕ s1 be a finite dimensional Lie superalgebra over k, see [40,Chapter 1]. The universal enveloping algebra U = U (s) of s is a Z 2 -graded associative algebra and a finite ring extension of U = U (s0).
In the following we use the term simple U -module to denote any of the following two notions (I) A simple U -module, without any reference to the Z 2 -grading; (II) A Z 2 -graded U -module which has no proper graded submodules.

Proposition 22
Assume that S ⊗ E has finite type socle, for every simple U -module S and any E ∈ F . Then every simple U -module is a quotient of a module of the form U ⊗ U L, for some simple U -module L.
Proof Consider a simple U -module K in the sense of (I). In particular, it is generated by any vector in K . We denote by Res the restriction functor from U -modules to U -modules, with left adjoint Ind and right adjoint Coind. Since Res(K ) is finitely generated, it is a noetherian Umodule. Consequently, there is a simple U -module S with non-zero morphism Res(K ) → S. By adjunction, we have an inclusion K → Coind S. Applying the exact restriction functor gives Since the right-hand side has finite type socle, there is a simple U -module L, for which we have an inclusion L → Res(K ). Applying adjunction shows that we have a surjection Ind L K . Consider a simple U -module K in the sense of (II). In particular, it is generated by any vector in K0 ∪ K1. We can follow the above procedure and make sure all relevant morphisms respect the Z 2 -grading. Any such morphism to or from a simple graded U -module will then again automatically be surjective or injective, respectively. The claim follows.

Semisimple Lie algebras over C
In this section, we work under the assumptions that k = C and that the Lie algebra g is semisimple.

Theorem 23
If g is of type A, then every module S ⊗ E, for S simple and E finite dimensional, has finite type socle.
Proof Let S be an arbitrary simple module with central character χ. Consider an indecomposable projective functor F such that F(S) = 0 and for which every indecomposable projective functor G F with G(S) = 0 satisfies G ∼ F. Note that this is possible since we only have finite chains We consider the equivalence class generated by F and take μ ∈ h * as in Lemma 14.
Since, by assumption, F ∼ Id η(μ) , we have projective functors Since F 1 S has finite type radical, we can choose a simple quotient L of F 1 S. By adjunction, we find that S is a submodule of G 1 L, for G 1 the right adjoint of F 1 .
By Theorem 21(ii), it thus suffices to prove that H L has finite type socle, for each projective endofunctor H of U -mod η(μ) . We claim that H L = 0 for every indecomposable projective endofunctor H different from the identity, which would thus complete the proof. Indeed, we even have H F 1 S = 0, since the statement in Lemma 14(ii),

Low ranks in Type B and C
, see the proof of [34, Theorem 31].
As in the proof of Theorem 23 it suffices to prove the following claim. Let L be a simple module in U -mod η(μ) which is annihilated by all projective endofunctors which are not in the equivalence class of Id η(μ) , then H L has finite type socle for each projective endofunctor H of U -mod η(μ) . In those cases where Id η(μ) is alone in its class there is nothing to prove. So we assume that there exists a second functor F in the class as in the above paragraph. However, in this case F 2 acts as the identity on simple modules which are annihilated by indecomposable projective endofunctors of U -mod η(μ) not in the class of Id η(μ) (in fact, F is an equivalence between appropriate subcategories of modules). Consequently F L is a simple module.

Remark 25
We can use the same arguments as in the proof of Theorem 24 for semisimple Lie algebras g = k ⊕ l, with k of type A and l of type B n or C n , for 1 ≤ n ≤ 4. Note that the arguments fail when we consider k ⊕ l, with both k and l of low rank type B and C.

Regular central character reduction for semisimple Lie algebras
Now we return to arbitrary semisimple Lie algebras over C. We will show that for Q2, it will suffice to consider simple modules with regular central character.

Remark 27
Theorem 26 remains true if χ r is also singular with W λ ⊂ W μ if we set n = |W μ |/|W λ |.
We keep the notation and assumptions of Theorem 26 for the rest of the section and start the proof with the following lemma. We freely use Lemma 12(ii), namely biadjunction between θ on and θ out and θ on θ out = Id ⊕n , freely.

Lemma 28
For a simple module L with central character χ s and a non-zero submodule or quotient N of θ out L, we have θ on N ∼ = L ⊕k with 1 ≤ k ≤ n. Moreover, for any simple module S, we have Hom g (S, θ out L) ∼ = Hom g (θ out L, S).
Proof We consider a quotient N , the proof for submodules being identical. By exactness of θ on , we find that θ on N is a quotient of L ⊕n . Since we have a non-zero morphism θ out L → N , we have a non-zero morphism L → θ on N , so we find θ on N = 0.
By the above, both sides of the proposed isomorphism are zero unless θ on S is semisimple. In the latter case, we have

If M is simple, then this monomorphism is an isomorphism.
Proof Take a natural transformation α : θ out ⇒ θ out . Then θ on (α) is a natural transformation Id ⊕n ⇒ Id ⊕n . By Lemma 13, we have End(Id ⊕n ) = Mat n (End(Id)), with End(Id) consisting only of scalar multiples of the identity natural transformation Id ⇒ Id. Hence a non-zero element in End(Id ⊕n ) evaluated at a non-zero module always yields a non-zero morphism. We claim that θ on (α) is not zero. It then follows that θ on (α) evaluated at any module M in U -mod 0 χ s is not zero, so in particular α M is not zero. To conclude the proof of injectivity, we can therefore just observe that we have θ on (α μ ) = 0. By Lemma 13, α μ : P λ → P λ is not zero and n = [P λ : L λ ]. It follows from [27, 4.12(3)] that θ on (β) = 0, for any non-zero endomorphism β of P λ . Now, let M = L be simple. In this case dim End g (θ out L) = dim Hom g (L, L ⊕n ) = n.
Hence, by Lemma 13 the dimensions of End g (θ out L) and End(θ out ) agree and the monomorphism must be an isomorphism.

Alternative proof
We view θ out as a functor from U -mod 0 χ s to the full subcategory C of Umod χ r of modules isomorphic to modules of the form θ out N , with N in U -mod 0 χ s . By construction, for this interpretation of θ out , End(θ out ) coincides with the algebra of natural transformations as in Lemma 13. Now the restriction of θ on to C has image contained in U -mod 0 χ s . Moreover, it also follows that (θ on , θ out ) is still a pair of bi-adjoint functors, in the above interpretation. Now adjunction and evaluation yields a commutative diagram End g (θ out M) ∼ Hom g (M, M ⊕n ).
By construction, we have to interpret Nat(Id, Id ⊕n ) as natural transformations with Id viewed as an endofunctor of U -mod 0 χ s . However, this does not differ from the space of natural transformations as in Lemma 13. Hence, dim Nat(Id, Id ⊕n ) = n and the space just consists of linear combinations of identity natural transformations of Id. In particular, the right vertical arrow is a monomorphism. This forces the left vertical arrow to be a monomorphism as well.
If M is simple, the right vertical arrow is, clearly, an isomorphism. Hence we find k = 1 and [θ out L : S] = n.

Proof of Theorem 26
Take some simple module S ∈ sim s r , then θ on S is non-zero and has finite type radical. In particular, there exists a simple module L with non-zero morphisms θ on S → L and S → θ out L. By Lemma 30, we have t(S) = L. Also by Lemma 30, the map t is a bijection and parts (i) and (iii) follow. Now we prove part (ii). By Theorem 16 it is sufficient to prove that every submodule of θ out L contains a simple subquotient of Gelfand-Kirillov dimension GK(L). For a submodule N ⊂ θ out L, Lemma 28 implies there exists a non-zero morphism L → θ on N . By adjunction, there is a non-zero morphism θ out L → N . Since θ out L has finite type radical and all simple quotients have Gelfand-Kirillov dimension GK(L) by Lemma 17, we find that N contains a simple subquotient of Gelfand-Kirillov dimension GK(L).
For a simple U -module S, we denote by S ⊗ F the full subcategory of the category of all g-modules consisting of all modules isomorphic to the ones of the form S ⊗ E, with E ∈ F .

Proposition 31 If, for every simple module S with regular central character, all modules in S ⊗F have finite type socle (resp. a GK-complete filtration), then, for every simple module L, all modules in L ⊗ F have finite type socle (resp. a GK-complete filtration).
Proof By Theorem 26, for every simple module L, there exists a simple module S with regular central character and V ∈ F such that L is a direct summand of S ⊗ V .
Hence each module in L ⊗ F is a direct summand of a module in S ⊗ F . The conclusion for socles follows from observing that any submodule of a module with finite type socle has finite type socle. The conclusion about GK-complete filtrations follows from Lemmata 17 and 10.

Corollary 32
Let w 0 be the longest element of a Coxeter subgroup W ⊂ W and S a simple module with integral regular central character. If θ w 0 S = 0, then θ x S has finite type socle, for all x ∈ W .
Proof We fix some dominant and integral μ such that W μ = W . Then we can write θ w 0 = θ out θ on using the notation as in Theorem 26. From Theorem 26 it follows that θ w 0 S has finite type socle and S = soc(θ w 0 S). From Kazhdan-Lusztig combinatorics it follows that θ x θ w 0 is a direct sum of copies of θ w 0 . Hence θ x S is a submodule of a direct sum of copies of θ w 0 S and also has finite type socle.

Direct sums of copies of sl 2
In this section, we fix j ∈ Z >0 and set g = sl 2 (C) ⊕ j . For this g, we can strengthen Theorem 23 as follows.
Theorem 33 For g = sl 2 (C) ⊕ j and any simple g-module S and E ∈ F , the module S ⊗ E has a GK-complete filtration. Moreover, if j = 1, then S ⊗ E even has finite length.
We label the j copies of sl 2 by the index 1 ≤ i ≤ j. We denote by i ∈ h * the weight half the positive root of the ith copy. Accordingly, we write λ = λ i i , with λ i ∈ C, for λ ∈ h * . The Weyl group acts on h * by changing signs for the λ i . For 1 ≤ i ≤ j, we denote by V i ∼ = C 2 the natural module for the ith copy of sl 2 , interpreted as a module for g in the obvious way. So, in this way we have supp Fix i such that 1 ≤ i ≤ j. Denote by mod i the full subcategory of U -mod consisting of all modules M on which the ith copy of sl 2 ⊂ g acts locally finitely. Clearly, − ⊗ V i restricts to an endofunctor on mod i . We also denote by sim i the set of isomorphism classes of simple g-modules which are not in mod i .

Lemma 34
Take a simple module S ∈ sim i . Then we have S 1 , S 2 ∈ sim i and a filtration of If j = 1, so g = sl 2 , the module S ⊗ V i has finite length.
Proof Let χ be the central character of S. We take a dominant λ ∈ η −1 (χ), which means and Lemmata 11 and 12(i) that − ⊗ V i restricted to U -mod χ decomposes into a direct sum of an equivalence with U -mod η(λ+ i ) and U -mod η(λ− i ) . Hence S ⊗ V i is a direct sum of two simple modules. That both are in sim i follows by adjunction.
If λ i ∈ Z >1 , the argument of the previous paragraph still applies. If λ = 1, we still have Eq. (5.1). We write θ on = F λ− i λ , as in Sect. 4.3. By Lemmata 11 and 12(i), S ⊗ V i is a direct sum of a simple module and θ on S. By Theorem 26(i), the latter module is zero or simple.
Finally, we consider λ i = 0, which implies We write θ out = F λ− i λ as in Sect. 4.3. Lemma 11 implies that S ⊗ V i ∼ = θ out S. That S ⊗ V i is of the desired form then follows from Theorem 26(i).
Since θ on θ out = Id ⊕2 , it follows that θ on F 1 T /F 2 T = 0. For g = sl 2 , Eq. (2.2) thus implies that F 1 T /F 2 T must be a direct sum of a number of copies of one fixed simple finite dimensional module. Since T is noetherian, it follows that F 1 T /F 2 T is finitely generated and thus finite dimensional.

Lemma 35 Let S be a simple U -module in mod i . Then S ⊗ V i is a direct sum of simple modules in mod i .
Proof Let χ be the central character of S. The explicit description of the generator of the center in the universal enveloping algebra of sl 2 implies that any dominant λ ∈ η −1 (χ) satisfies λ i ∈ Z >0 . It then follows from the same arguments as in Lemma 34 that S ⊗ V i is simple, if λ i = 1, and is a direct sum of two simple modules if λ i > 1.

Corollary 36
For any simple g-module S and 1 ≤ i ≤ j, the module S⊗V i has a GK-complete filtration.
Proof By Lemma 35, we can assume S ∈ sim i .
We fix an arbitrary d ∈ N. First we define an equivalence relation on simple g-modules in sim i with Gelfand-Kirillov dimension d. We let ∼ be the minimal equivalence relation generated by relation S ∼ S if we have a non-zero morphism S ⊗ V i → S or S → S ⊗ V i . Since V i is self-dual, this is indeed an equivalence relation. Take a simple module S with GK(S) = d and e(S) minimal in its equivalence class. Set e := e(S). By Eq. (2.3), we have e(S ⊗ V i ) = 2e. The simple modules S 1 , S 2 in Lemma 34 thus satisfy e(S 1 ) + e(S 2 ) ≤ 2e by Lemma 8(ii). By minimality of e, we thus have e(S 1 ) = e = e(S 2 ). With notation of Lemma 34, it follows that we have GK(F 1 T /F 2 T ) < d. Consequently S ⊗ V i has a GK-complete filtration.
The above paragraph also shows that S 1 and S 2 are the unique simple modules which appear as submodules or quotients of S ⊗ V i , by Lemma 17. Applying the same procedure to S 1 , S 2 iteratively shows that every simple module in the equivalence class of S has Bernstein number e(S). Hence the condition for S to have minimal e(S) in its equivalence class was not actually a restriction. This completes the proof.

Proof of Theorem 33
Every finite dimensional g-module is a direct sum of a module of the form The statement about GK-complete filtrations thus follows from Corollary 36. The statement for socles then follows from Corollary 20.
The claim for sl 2 follows from Lemma 34.

Setup
In this section, we will introduce the setup of Lie superalgebras. We refer to [12,40] for more details. From now on we work over the field C of complex numbers. We let g be a finite-dimensional Lie superalgebra with Z 2 -graded decomposition From now on we assume that g := g0 is a reductive Lie algebra of type A and that g1 is a semi-simple g0-module. The Weyl group W of g is defined to be the Weyl group of the reductive Lie algebra g and we keep notation and terminology of Sect. 2.4. Let V = V0 ⊕ V1 be a superspace. For a given homogeneous element v ∈ V i , where i ∈ Z 2 , we let v= i denote its parity. We denote the parity change functor by on the category of superspaces, cf. [12, Section 1.1.1]. For a Z 2 -graded associative algebra A, we denote by A-smod the category of all finitely generated Z 2 -graded modules with grading preserving homomorphisms. Note that when A is reduced, i.e. A = A0, we have

Categories of (super)modules
We denote the universal enveloping algebras of our Lie (super)algebras by U := U ( g) and U := U (g) = U ( g0). Let Z ( g) and Z (g) denote the center of U and U , respectively. Also, the center of g is denoted by z(g).
By [2, Theorem 2.2] (also see [21]), the functors Ind and Coind are isomorphic, up to the equivalence given by tensoring with the one-dimensional g-module on the top degree subspace of g1.
By abusing notation we let O be the full subcategory of U -smod of weight modules which are locally b-finite. In principle, we would have to write O ⊕ O for this category, in order to be consistent with Sect. 2.4. Similarly, we let L λ denote the simple g-module in unspecified parity. We denote by O the full subcategory of U -smod of supermodules which restrict to O.

Theorem 37
Let V be a simple g-supermodule. Then there exist λ ∈ h * such that Ann U (V ) = Ann U (L λ ).
We let F denote the category of finite dimensional semisimple g-supermodules, and we let F denote the category of finite-dimensional g-supermodules which restrict to objects in F . Then F and F are exactly the categories of finite-dimensional weight g-supermodules and g-supermodules, respectively.
For a g-supermodule M, we denote by We let Coker( F ⊗ M) denote the coker-category of M, that is the full subcategory of the category of all g-supermodules, which consists of all modules N that have a presentation Similarly we define analogous full subcategories of gsupermodules, cf [38].

Harish-Chandra bimodules
Following [9, § 3.2], we will consider a type of Harish-Chandra bimodules where the left action is by a Lie superalgebra and the right action by the underlying Lie algebra. The corresponding category will be an essential tool in our study of the rough structure.
We write U -smod-U for U ⊗ U op -smod. For a bimodule Z in this category, the g-module Z ad is the restriction of Z to the adjoint action of g. This is the restriction via Let B denote the full subcategory of U -smod-U of bimodules N for which N ad is a direct sum of modules in F with finite multiplicities. For a two-sided ideal J ⊂ U , we let B(J ) denote the full subcategory of B of bimodules X such that X J = 0. We have the functor N ) is the maximal submodule of Hom C (M, N ) which belongs to B. By slight abuse of notation, we will use the same notation L for the corresponding functor applied to the case g = g. Let M be a U -module, then the g-action on M defines a bimodule homomorphism from U to L ( For E ∈ F , we recall that E ⊗ U is equipped with a natural g-bimodule structure as in [3, Section 2.2] and [15, Section 2.4]: for all homogeneous X , Y ∈ g, v ∈ E and u ∈ U . 7 Rough structure of simple g-supermodules

Motivation
Two fundamental problems in representation theory of a group or a ring are: • classification of simple modules; • understanding how all modules are constructed from simple modules.
A natural subproblem of the second problem is determination of multiplicities of simple subquotients in a given module. In the general case, certain multiplicities might be infinite or depend on the choice of filtration.
The paper [30] studied the structure of a certain class of modules over Lie algebras, called generalized Verma modules. These modules are obtained by parabolic induction from simple modules over "smaller" Lie algebras. It turned out that, given a generalized Verma module, there is a natural class of simple modules (defined using a certain comparability of annihilators) whose multiplicities in the generalized Verma module are well-defined, finite and computable using Kazhdan-Lusztig combinatorics. This was called the rough structure of generalized Verma modules. The most general, to date, result on the rough structure of generalized Verma modules was obtained in [38,Section 11.8].
In the study of Lie superalgebras, it is natural to ask about the rough structure of simple supermodule as modules over the even part of the Lie superalgebras. In [10], for classical Lie superalgebras of type I , it was shown that • the classification of simple supermodules can be reduced to the classification of simple modules over the even part of the superalgebra; • the rough structure of a simple supermodule as a module over the even part of the Lie superalgebra can be described in terms of combinatorics of category O.
The goal of this section is to address these problems for Lie superalgebra not necessarily of type I, but with underlying Lie algebra of type A.

Coker categories for induced modules
With Theorem 38 as the main tool, this subsection shall proceed with the study of cokercategories of induced modules along the lines of [38,Section 11.6].
For simplicity, we will work with regular integral central characters by following [38,Remark 76]. The general case then follows by standard techniques, in particular using translations out and onto the walls and the equivalences from [14].
We start with reviewing and adapting the setup of [38,Section 11.6] to the present paper. Note that we work in the generality when g is reductive. Each central element of g acts as a scalar on any simple g-module, therefore we will work only with g-modules on which the action of the center of g is semi-simple. We assume that the center of g belongs to any Cartan subalgebra and we call a weight integral if it appears in some simple finite dimensional g-module.
Let L be a simple g-supermodule which admits a regular and integral central character. By Theorem 37, there is a dominant weight ν and an element σ ∈ W such that Ann U (L) = Ann U (L σ ν ). We may assume that σ is contained in a right cell associated with a parabolic subalgebra p ⊆ g as in [38,Remark 14]. Therefore there is a dominant weight μ such that the parabolic block O p μ contains exactly one simple module L yμ and this module is projective, see e.g. [26, 3.1]. As yμ = yxμ, for any x in the stabilizer of μ, without loss of generality, we may assume that ys < y, for all simple reflection s with sμ = μ. With this assumption, we have that L yμ is the translation of L y0 to the μ-wall.
Tensoring, if necessary, with finite dimensional modules, without loss of generality we may assume that μ is generic in the sense of [37,Subsection 5.3]. Let F be a projective functor given in [38,Proposition 62]. A simple top of F L is defined uniquely, up to equivalence given by the action of projective functors. We fix one representative N in such an equivalence class. We refer the reader to [38,Section 11] for more details of our setup. In particular, we have that The following theorem, which is [38,Theorem 66], is our main tool to study the rough structure for Lie algebras of type A.

Theorem 39
Let F denote the category of finite-dimensional weight g-supermodules. Then the functor is an equivalence of categories.

Lemma 40
The objects L yμ and N are projective in F ⊗ L yμ and F ⊗ N , respectively. Now we can formulate the following equivalence of coker-categories.

Corollary 41
The functor is an equivalence sending Ind(N ) to Ind(L yμ ).
Proof With I from 7.1, we claim that as U -modules. This completes the proof.

Rough structure of induced g-supermodules
In this subsection, we obtain a description of the rough structure of induced supermodules.

Lemma 42
The equivalence in Corollary 41 gives rise to a bijection between the sets of isomorphism classes of objects in the categories add( F ⊗ Ind(N )) and add( F ⊗ Ind(L yμ )). These two categories are, respectively, the categories of projective objects in Coker( F ⊗ Ind(N )) and Coker( F ⊗ Ind(L yμ )).
Proof For a given E ∈ F , we have Here the first and the last isomorphisms use [31,Proposition 6.5], which states that for any g-module X , the second isomorphism uses [27,Section 6.8]  As a consequence, see [38,Section 11], induces a bijection : Irr g (Coker( F ⊗ Ind(N ))) → Irr g (Coker( F ⊗ Ind(L yμ ))), between the sets of isomorphism classes of simple g-supermodule quotients of simple objects in Coker( F ⊗ Ind(N )) and Coker( F ⊗ Ind(L yμ )).
For S ∈ Irr g (Coker( F ⊗ Ind(N ))), we define We are now in a position to state the first main result of this section which describes rough structure of induced modules in terms of category O combinatorics.
Theorem 43 (Rough structure of induced modules) For any module E ∈ F and any module S ∈ Irr g (Coker( F ⊗ Ind(N ))), we have The proof of Eq. (7.4) follows from a similar argument using as given in [38,Theorem 72]. To see this, let Q S and Q L denote the indecomposable projective objects in the categories Coker( F ⊗ Ind(N )) and Coker( F ⊗ Ind(L yμ )) with tops S and L S , respectively. By (7.3) we have The claim follows.

Rough structure of simple g-supermodules
In this subsection, we give a description of the rough structure of restrictions to g of simple g-supermodules in Irr g (Coker( F ⊗ Ind(N ))).

Lemma 44
There is an isomorphism Res • ∼ = • Res of functors.
Proof This follows directly from the definitions.
We have a bijection : induced by , between the sets of isomorphism classes of the simple g-quotients of simple objects in Coker(F ⊗ N ) and in Coker(F ⊗ L yμ ). For a given we define the related weight ζ W ∈ h * by L ζ W ∼ = (W ). The next statement describes the g-rough structure of simple g-supermodules in terms of the combinatorics of category O.

Classification of simple g-supermodules
Theorem 39 reduces the problem of classification of all simple g-modules to the following problem. Let I denote the set of all primitive ideals in U which appear as annihilators of all modules of the form N , as chosen before (7.1). For each fixed simple g-module S with annihilator I ∈ I (which we view as some N ), we have the corresponding bijection (S) : Irr g (Coker(F ⊗ S)) → Irr g (Coker(F ⊗ L yμ )).
For I ∈ I, let Irr g (I ) denote the set of isomorphism classes of simple g-modules whose annihilator is I . Consequently, we have:

Corollary 46
The set coincides with the set of isomorphism classes of simple g-modules.
This reduces, in some sense, the problem of classification of all g-modules to that of classification of all g-modules with annihilators in I. For Lie algebras, this might look unimpressive. However, a similar reasoning applied to Corollary 41 gives the following very surprising extension of this to g which, in the same sense, reduces the problem of classification of all g-supermodules to that of classification of all g-modules with annihilators in I. Here, again, for each fixed simple g-module S with annihilator I ∈ I, we have the bijection (S) : Irr g Coker( F ⊗ Ind(S)) → Irr g Coker( F ⊗ Ind(L yμ )) .
Theorem 47 (Classification of simple supermodules) The set I ∈I S∈Irr g (I ) Irr g Coker( F ⊗ Ind(S)) (7.6) coincides with the set of isomorphism classes of simple g-supermodules.
Proof As g is of type A, we can combine Theorem 23 with Proposition 22 to conclude, using Corollary 41, that every simple g-supermodule appears in the right hand side of (7.6). The claim follows.
Although Theorem 47 is not as nice as [10, Theorem 4.1] (the latter result reduces classification of simple supermodules over basic classical Lie superalgebras of type I to classification of simple modules over the corresponding even part Lie algebra), it is fairly clear that [10, Theorem 4.1] does not extend to, for example, Q-type Lie superalgebras given in Subsection 7.6 in any easy way. At the same time, the set Irr g Coker( F ⊗ Ind(L yμ )) for Q-type Lie superalgebras can be described using Sect. 7.6. Therefore for Q-type Lie superalgebras, Theorem 47 provides significant progress in classification of simple supermodules.

Applications
We give a short overview of Lie superalgebras to which our results are applicable and cannot be dealt with using the theory of [10]. We focus on D(2, 1; α), F(4), Q(n) and generalized Takiff superalgebras.
First we quickly review the classification of simple modules in O. Let g be a classical Lie superalgebra and choose a Cartan subalgebra a of the derived Lie algebra g = [g, g] of the underlying Lie algebra. Consider a Weyl group invariant bilinear form ·, · on a * . Following [35, § 2.4], we choose a ω ∈ R , where is the set of roots of g , which is generic in the sense that ω, α = 0 for all α ∈ . Then we define a triangular decomposition where h is the direct sum of a-weight spaces corresponding to weights λ satisfying λ, ω = 0 and n ± corresponds to weights λ with ± λ, ω > 0. Clearly h := h0 and n ± := n ± 0 give a triangular decomposition of g.
Assume now that [ h1, h1] = 0. Then simple h-modules are the one dimensional h-modules with trivial h1-action C λ , with λ ∈ h * . We have the corresponding Verma module with b = h ⊕ n + acting on C λ with trivial n + -action. It follows as in the classical case that λ has a unique maximal submodule. We denote the quotient by L λ . The following lemma is well-known, see e.g. [35,Proposition 2].

Lemma 48
The set { L λ , L λ | λ ∈ h * } is a complete and irredundant set of representatives of isomorphism classes of simple objects in O.
The one parameter family of simple Lie superalgebras D(2, 1|α), see [40,Chapter 4], satisfies the above assumptions and has underlying Lie algebra of type A. The multiplicities in Res( L λ ) for D(2, 1|α) have been determined in [13]. Therefore our results provide a concrete approach to the rough structure problem. The simple Lie superalgebra F(4), see [40,Chapter 4], has underlying Lie algebra so 7 ⊕ sl 2 . Based on Remark 25 we can thus expect to apply some of our results to this case. Note that there are also many examples of algebra g as above which are far from simple, see for instance the generalized Takiff superalgebras introduced in [23, Section 3.3].
For Lie superalgebras of type Q, see [12,40], the statement in Lemma 48 has to be adapted, in particular, as some simple highest weight supermodules in type Q are invariant under the parity change. However, an explicit classification of simple highest weight modules is known, see e.g., [12,Section 1.5.4] and [11, Lemma 2.1]. The category O for type Q Lie superalgebras has been intensively studied in e.g. [6,7,18,19]. However, the results in type Q are not yet as complete as, say, for gl(m|n). We conclude this subsection with more detailed analysis of two small rank examples for type Q superalgebras: Example 49 For the Lie superalgebra q 1 , we have that U (q 1 ) is isomorphic to C[x], where x is odd. In particular, each simple supermodule is finite dimensional and thus belongs to O. The set I is a singleton and we can choose it to consist of any fixed primitive ideal I . The set Irr g (I ) is also a singleton. In this case the classification provided by Theorem 46 is tautological.
Example 50 For the Lie superalgebra q 2 , a classification of simple supermodules can be found in [36] where it is given in a slightly different language and is much more detailed.
Namely, [36] defines a correspondence between primitive ideals of U (q 2 ) and those of U (gl 2 ) and constructs explicit U (q 2 )-U (gl 2 )-bimodules, tensoring with which provides a bijection between simple U (gl 2 )-modules and simple U (q 2 )-supermodules with the annihilators connected by the defined correspondence. The description in Theorem 46 can be obtained from the description in [36] by putting together the classification pieces from [36] for primitive ideals connected by tensoring with finite dimensional (super)modules. In the q 2 case, there are two essentially different types of primitive ideals: those of finite codimension, for which Theorem 46 is tautological, just as in Example 49; and those of infinite codimension. In the first case, each Irr g (I ) is a singleton. In the second case, a classification of the elements in each Irr g (I ) can be obtained from the results of [5]. The set I can be chosen to consist of the annihilator of the trivial module, the annihilator of one fixed singular Verma module and the annihilator of one fixed simple Verma module in each weight lattice that does not contain any singular Verma modules.

Kac functor preserves finite type socle and finite type radical for Lie superalgebras of type I
In this section we prove the following observation which naturally connects to various previous parts of the paper. To state the most general result we work in the category g-sMod of all supermodules.
Theorem 51 Let g = g −1 ⊕ g 0 ⊕ g 1 be a Lie superalgebra of type I with g := g 0 and p := g 0 ⊕ g 1 . Then the corresponding Kac functor K := Ind g p : g-sMod → g-sMod (where g 1 acts trivially on g-modules) sends modules with finite type socle (resp radical) to modules with finite type socle (resp. radical) preserving the length of the socle (resp. radical).
Proof We start with the claim about the socle. Let V ∈ g-sMod be a module with finite type socle. Set d := dim g −1 . Any g-submodule of K(V ) intersects d g −1 ⊗ V , see [10, Lemma 3.1]. The space d g −1 ⊗ S, with S a simple submodule of V , therefore generates a simple g-module of K (V ) and moreover intersects d g −1 ⊗ V precisely in d g −1 ⊗ S. This follows from considering K (V ) as a Z-graded module where elements of g ±1 act as operators in degree ±1. The submodule generated by d g −1 ⊗ soc(V ) is thus semisimple, essential and has same length as soc(V ). We proceed with the claim about radical. By Remark 5(i) and (iii) it follows that K sends modules with finite type radical to modules with superfluous radical. For an arbitrary g-module M and g-module N , we have Hom g (K (M), N ) ∼ = Hom g (M, N g 1 ) with N g 1 = {v ∈ N : g 1 v = 0}.
The above equation and [10, Theorem 4.1(i)] imply that S L := L g 1 is a simple g-module for each simple g-module L, and the map L → S L induces a bijection of isomorphism classes of simple modules. For an arbitrary g-module M and a simple g-module L we thus have Consequently, if M has finite type radical, the length of the top of M and K (M) coincide.
As an immediate corollary, we obtain the following. We call a Borel subalgebra b of g distinguished if it is of the form b ⊕ g 1 , for a Borel subalgebra b of g.

Corollary 52 All Verma supermodules with respect to distinguished Borel subalgebras over all Lie superalgebras of type I have simple socle.
Proof Verma supermodules over Lie superalgebras of type I can be obtained from Verma modules over Lie algebras using Kac functor. Therefore the claim follows from Theorem 51 and [16, Proposition 7.6.3(i)].