Classification of Simple Harish-Chandra Modules for Basic

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can be seen, for example, in the study of the category of finite-dimensional modules over a finite-dimensional simple Lie superalgebra, which is not semisimple in general [26]; or in the study of the primitive spectrum of a universal enveloping superalgebra [13]. In this paper, we consider classes of Lie superalgebras that have drawn a lot of attention in recent years. Namely, the classes of map superalgebras and affine (Kac-Moody) Lie superalgebras. Given an affine scheme of finite type X and a Lie superalgebra g, both defined over the same ground field k, the map superalgebra associated to X and g, denoted by M(X, g), is the Lie superalgebra of all regular maps from X to g. It is useful to note that the map superalgebra M(X, g) is isomorphic to the tensor product := g k A, G ⊗ where A = X (X). Map superalgebras provide a unified way of realizing many important classes of LieO superalgebras, such as generalized current superalgebras, loop and multiloop superalgebras, Krichever-Novikov superalgebras. The affine Lie superalgebra (g) and the affine Kac-Moody Lie superalgebra g are obtained via certain extensions ofL the loop superalgebra M(C×, g). Representation theory of map algebrasb is fairly well developed. The classification of all simple finite-dimensional modules was first obtained for loop algebras by Chari and Pressley [7, 14], and then generalized for arbitrary map algebras [28]. In the super setting, a combination of results from [30, 12, 10] provides a classification of all simple finite-dimensional modules over any classical map superalgebra. In both, super and non- super cases, the classification relies on a class of modules called (generalized) evaluation modules, which will be also important in this paper (see Section 4). Other classes of finite-dimensional modules were also studied for map algebras and superalgebras in [21, 6, 22, 1, 9]. A comprehensive program seeking a description of simple Harish-Chandra modules (or finite weight modules as we call them throughout the text), that is, simple weight modules with finite-dimensional weight subspaces over a Lie algebra started at the end of the 1980’s with works of Fernando, Britten and Lemire [20, 2, 3]. The classification of all such modules over finite-dimensional reductive Lie algebras was then obtained by Mathieu in [29]. Similar approaches were also developed for finite-dimensional and affine Kac-Moody Lie (super)algebras in [16, 23, 17, 19, 24, 5, 33], and for map algebras over finite-dimensional reductive Lie algebras in [4, 27]. In the current paper, we make a significant step in the study of infinite-dimensional modules over map superalgebras and in the classification of simple Harish-Chandra modules over affine (Kac-Moody) Lie superalgebras. Namely, we classify all simple Harish- Chandra -modules, under the assumption that g is a basic classical Lie superalgebra. We proveG that any such module is either cuspidal bounded, or parabolic induced from a cuspidal bounded module over a cuspidal subalgebra of . Moreover, we show that any cuspidal bounded module is isomorphic to an evaluation module.G This classification yields in turn a classification of the same class of modules over the affine Lie superalgebra (g). Hence, our first main result is the following. L

Theorem 0.1. (i) Every simple Harish-Chandra -module is parabolic induced from a simple bounded cuspidal module over the LeviG factor; (ii) If g0 is semisimple (in particular, this is the case when admits cuspidal modules), then any simple cuspidal bounded -module is isomorphicG to an evaluation module. G CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 3

Moving to affine Kac-Moody Lie superalgebras, we construct a family of bounded simple modules over g generalizing the results of [14], and conjecture that all bounded simple cuspidal g-modules belong to this family. Our second main result is the following. ∗ Theorem 0.2.b Let λ1,...,λk h , F (λ1),...,F (λk) the corresponding simple highest ∈ × weight g-modulesb and V0 a simple weight g-module. If a0, a1,...,ak C such that r −∈r ai = aj for i = j, and there is λ0 Supp V0 for which im χ(λ,a) = C[t , t ], then we have an6 isomorphism6 of g-modules ∈ r−1 a0 a1 ak i a0 a1 ak L(V0 F (λb1) F (λk) ) = L (V0 F (λ1) F (λk) ) ⊗ ⊗···⊗ ∼ ⊗ ⊗···⊗ i=0 M where each Li(V a0 F (λ )a1 F (λ )ak ) is a simple g-module. 0 ⊗ 1 ⊗···⊗ k We observe that in the case of Lie algebras, these are exactly the simple modules that appear in the unpublished work of Dimitrov and Grantcharovb [15]. It is a well known fact that all simple finite-dimensional modules over g can be obtained as simple quotients of Kac modules. Moreover, it was shown in [19] that in some cases any simple Harish-Chandra module of g of nonzero level is parabolic induced from a cuspidal simple module over a subalgebra of even part of g, e.g. in the case of sl\(n m). | This phenomenon was studied in [11] forb general superalgebras with finite-dimensional odd part, in which case simple modules are obtained as simpleb quotients of Kac modules. For basic classical Lie superalgebras of type I, we use the Kac functor to reduce the classification of all bounded simple g-modules of level zero to the classification of the same class of modules over g0. Namely, we have the following result. Theorem 0.3. Let g be a basic classicalb Lie superalgebras of type I. Then the Kac induction functor gives a bijectionb between the sets of isomorphism classes of simple bounded Harish-Chandra g-modules and simple bounded Harish-Chandra g0-modules. This paper is organized as follows. In Section 1 we fix notation and state basic results that will be neededb in the subsequent sections. We prove someb results regarding tensor product of infinite-dimensional representations in Section 2. We stress that these results are quite interesting by their own, as they generalize (to the infinite-dimensional super setting) classical statements which we could not find in the existing literature in the generality we need. In Section 3 we prove a parabolic induction theorem for finite weight modules over map superalgebras associated to basic classical Lie superalgebras. The proof is based on the concept of the shadow of a module, which was introduced by Dimitrov, Mathieu and Penkov in [16], and reduces the classification problem to the problem of classifying all cuspidal bounded modules over cuspidal subalgebras of . In Section 4, we classify all cuspidal bounded modules over cuspidal map superalgebrasG in terms of evaluation modules. Section 5 is devoted the case of affine (Kac-Moody) Lie superalgebras. Firstly, we apply our results for the particular case of loop superalgebras M(C×, g) to obtain a description of all finite weight modules over (g). Secondly, we generalize results of [14] and we use these results along with the classificationL of simple bounded modules over M(C×, g) to construct a family of level zero simple bounded modules over g. In the last subsection we assume that g is a basic classical Lie superalgebras of type I in order to define analogs of Kac g-modules. The main result of this subsection generalizesb to the affine setting the reduction results from [19] and [11] for bounded modules. b 4 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

g g0 Type

A(m, n), m > n 0 Am An k I A(n, n), n 1 ≥ A ⊕A ⊕ I ≥ n ⊕ n C(n + 1), n 1 Cn k I B(m, n), m ≥ 0, n 1 B ⊕ C II ≥ ≥ m ⊕ n D(m, n), m 2, n 1 Dm Cn II F (4) ≥ ≥ A ⊕B II 1 ⊕ 3 G(3) A1 G2 II D(2, 1, a), a =0, 1 A ⊕ A A II 6 − 1 ⊕ 1 ⊕ 1 Table 1. Basic classical Lie superalgebras that are not Lie algebras, their even part and their type

1. Preliminaries In this paper, we fix an uncountable algebraically closed field k with characteristic 0. Let g = g0 g1 be a basic classical Lie superalgebra with Cartan subalgebra h g0 (see [25]). Let A⊕be a finitely generated associative commutative algebra with identity⊂ over k, and set = g A the map superalgebra associated to g and A. For any Lie superalgebra a we denoteG by U⊗(a) its universal enveloping superalgebra. A -module V is called a weight module if V is a weight module over g = g k , that is,G if we have a decomposition λ λ ∼ ⊗ ⊆ G V = ∗ V , where V = v V hv = λ(h)v for all h h is the weight space λ∈h { ∈ | ∈ } associated to the weight λ. The support of V is the set Supp V = λ h∗ V λ = 0 . L Throughout this article a -module is always assumed to be a weight{ module.∈ | A6 finite} weight module is a weightG module whose dimensions of all its weight spaces are finite. A finite weight module V is said to be bounded if there is n 0 for which dim V λ n for all λ Supp V . Notice that is a weight module under the≫ adjoint action, and α≤= gα A for∈ all α Supp . Thus G is a finite weight module if and only if dim A< G. Moreover,⊗ the root∈ systemG of is theG same as that of g and it is given by ∆ := Supp∞ 0 . In G G\{ } particular, we have that ∆ = ∆0 ∆1, and every root of is either in ∆0 or in ∆1. For α ∪ G α each α ∆, we have that dim g = 1, and hence we will fix a nonzero vector xα g , and we set h∈ := [x , x ]. When α ∆ , we will always assume x , x , h is a ∈sl -triple. α α −α ∈ 0 { α −α α} 2 We define = xα, hα α ∆ g, which generates g as a vector space. We will denote by = Z∆B (resp.{ =| Z∈∆ )} the ⊆ lattice generated by ∆ (resp. ∆ ). Q Q0 0 0 Recall that modules over g0 on g1 is either simple, in which case g is said to be of type II, or it is a direct sum of two simple modules, and we say g is of type I. In Table 1 we list all basic classical Lie superalgebra that are not Lie algebras, together with their even part and their type. Note that g0 is either semisimple or a reductive Lie algebra with one-dimensional center. Any basic classical Lie superalgebra g admits a distinguished

Z-grading g = n∈Z gn, which is compatible with the Z2-grading and satisfies: (1) if g is of type I, then g = g , g = g g , and L 0 0 1 −1 1 (2) if g is of type I, then g = g g g⊕, g = g g . 0 −2 ⊕ 0 ⊕ 2 1 −1 ⊕ 1 The analogue for Lie superalgebras of the Poincaré-Birkhoff-Witt Theorem (PBW The- orem) is formulated in the following way (see [31, Chapter §2, Section 3]). CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 5

Lemma 1.1. Let a be a Lie superalgebra over k. Let B0, B1 be totally ordered basis of a0, a1, respectively. Then the standard monomials u ...u v ...v , u ,...,u B , v ,...,v B ,u u , v < < v , 1 r 1 s 1 r ∈ 0 1 s ∈ 1 1 ≤···≤ r 1 ··· s form a basis for the universal enveloping superalgebra U(a). In particular, we have an isomorphism of vector spaces U(a) = U (a ) (a ) . ∼ 0 ⊗ 1 For any -module V , define ^ G Ann (V ) := a A (g a)V =0 . A { ∈ | ⊗ } The following results reduces the study of simple finite weight modules over an arbitrary map superalgebra to the study of simple finite weight modules over certain finite- dimensional map superalgebras. Proposition 1.2. [30, Proposition 8.1] If an ideal of , then exists an ideal I of A such that = g I. I G I ⊗ The proof of the following proposition is similar to the Lie algebra case. See [4, Propo- sition 4.3].

Proposition 1.3. If V is a simple ﬁnite weight -module, then A/Ann A(V ) is a ﬁnite- dimensional algebra. G

2. A tensor product theorem This section is devoted to prove an inﬁnite-dimensional version of a classical result proved by Cheng [8]. Namely, that if Vi is a simple module over a Lie superalgebra gi, for i =1, 2, then the (g g )-module V V is either simple or there is a simple submodule 1 ⊕ 2 1 ⊗ 2 V V1 V2 for which V1 V2 ∼= V V . We start with a generalization of Schur’s lemma to⊆ the super⊕ inﬁnite-dimensional⊗ setting.⊕ Lemma 2.1 (Schur’s Lemma). Let g be a Lie superalgebra and V be a simple g-module. Then one of the following statements hold:

(1) Endg(V ) = Endg(V )0 = kid, (2) V = V , and End (V )= kid kσ, where σ2 = id is an odd element that permutes 0 ∼ 1 g ⊕ V0 and V1. In particular, σ provides an isomorphism of g0-modules between V0 and V1.

Proof. We know that Endg(V ) = Endg(V )0 Endg(V )1, where Endg(V )0 = kid by the inﬁnite-dimensional Schur’s Lemma (due⊕ to Diximier) see [18, Problem 2.3.18]. If End (V ) = 0, then we have case (1). Suppose End (V ) = 0, and take a nonzero ele- g 1 g 1 6 ment σ Endg(V )1. Since ker σ is a proper g-submodule of V , we see that ker σ =0. On the other∈ hand, as im σ is a nonzero g-submodule of V , we have that im σ = V . In particular, for σ2 End (V ) we obtain σ2 = Kid for some nonzero K k. We can assume ∈ g 0 ∈ K = 1. Now let τ Endg(V )1 be a nonzero element. As before we get that σ τ = K1id for a nonzero K ∈k, which implies σ2 τ = id τ = τ = K σ id = K σ. ◦ 1 ∈ ◦ ◦ 1 ◦ 1 Lemma 2.2. Let g be a Lie superalgebra, and let V be a simple g-module. 6 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

(1) Assume that Endg(V )1 = 0, that v1,...,vn V are linearly independent vectors and that w ,...,w V are arbitrary elements.∈ Then there exists an element 1 n ∈ u U(g) such that uvi = wi. (2) Assume∈ that End (V ) = 0, that v ,...,v V are linearly independent vectors g 1 6 1 n ∈ i and that w1,...,wn Vi are arbitrary elements, where i 0, 1 . Then there exists an even element∈ u U(g) such that uv = w . A ∈ { } ∈ 0 i i Proof. Both statements follow from the Jacobson Density Theorem and Schur’s Lemma (see [8, Proposition 8.2]). Also observe that u is even in (2) since all vi’s and all wi’s have the same parity.

Lemma 2.3. Let g1 and g2 be Lie superalgebras, and V1, V2 be simple modules over g1 and g , respectively. If End V = k, then V V is a simple g g -module. 2 g1 1 ∼ 1 ⊗ 2 1 ⊕ 2 Proof. We need to show that for any nonzero v V V we have that U(g g )v = V V . ∈ 1⊗ 2 1⊕ 2 1⊗ 2 Let v V1 V2 be an arbitrary homogeneous nonzero element of V1 V2. Then we can write v∈= ⊗r v1 v2. Suppose, without loss of generality, that v1⊗r is a k-linearly j=1 j ⊗ j { j }j=1 independent set and v2 = 0. By Lemma 2.2 (1), there is u U(g ) such that uv1 = δ v1. P 1 1 j j1 1 Thus, u U(g g ) and6 ∈ ∈ 1 ⊕ 2 r uv = (uv1) v2 = v1 v2 =0. j ⊗ j 1 ⊗ 1 6 j=1 X Let w V and w V be arbitrary elements. By the irreducibly of V and V as 1 ∈ 1 2 ∈ 2 1 2 modules over g1 and g2, respectively, there exists u1 U(g1) and u2 U(g2) such that 1 2 ∈ ∈ u1v1 = w1 and u2v1 = w2. Thus, 1 2 u1((u2(uv))) = u1(u2(v1 v1)) ⊗ 1 |u2||v1| 1 2 = u1(( 1) v1 (u2v1)) − 1 ⊗ |u2||v1 | 1 =( 1) u1(v1 w2) − 1 ⊗ 1 =( 1)|u2||v1 |(u v1) w =( 1)|u2||v1|w w . − 1 1 ⊗ 2 − 1 ⊗ 2 Therefore, w1 w2 U(g1 g2)v for all w1 w2 V1 V2, which implies that V1 V2 = U(g g )v. ⊗ ∈ ⊕ ⊗ ∈ ⊗ ⊗ 1 ⊕ 2 Proposition 2.4. Let g1 and g2 be Lie superalgebras, and V1, V2 be simple modules over g1 and g2, respectively. Then V1 V2 is either a simple g1 g2-module, or it is isomorphic to V V , where V is a simple g⊗ g -module. ⊕ ⊕ 1 ⊕ 2 k k Proof. If Endg1 (V1) ∼= or Endg2 (V2) ∼= , then by Proposition 2.3 V1 V2 is a simple g g k k ⊗ 1 2-module. By Lemma 2.1 we can assume that Endg1 (V1) ∼= id σ1 and Endg1 (V1) ∼= kid⊕ kσ , where σ and σ are odd elements such that σ2 = id and⊕ σ2 = id. ⊕ 2 1 2 1 2 − We have that σ = σ1 σ2 is a g1 g2-module endomorphism. Since σ(v1 v2) = ⊗ 2 ⊕ ⊗ σ1(v1) σ2(v2), we see that σ = id. Note⊗ that for every x V V ∈ 1 ⊗ 2 x σ(x) x + σ(x) x = − + , 2 2 CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 7 thus V V = V V ′ with 1 ⊗ 2 ⊕ V = x V V σ(x)= x and V ′ = x V V σ(x)= x . { ∈ 1 ⊗ 2 | } { ∈ 1 ⊗ 2 | − } Note that V and V ′ are g g -submodules of V V . Let w V i I be a basis 1 ⊕ 2 1 ⊗ 2 { i ∈ 2 | ∈ } of the even part V2, then wi, σ2(wi) i I is a basis of V2. It is possible to show that { ∗ | ∈ } ′ V is generated by v wj + σ1(v) σ2(wj) v V1, j I , and V is generated by ∗ { ⊗ ⊗ | ∈ ∈ } v wj σ1 (v) σ2(wj) v V1, j I . { ⊗ − ⊗ | ∈ ∈ } ′ The automorphism of g1 g2-modules σ1 1 : V1 V2 V1 V2 sends V to V and ′ ′ ⊕ ⊗ ⊗ → ⊗ V to V , thus V ∼= V . Let v V be a homogeneous element of V . Then, ∈ k v = u w + σ∗(u ) σ (w ), ir ⊗ ir 1 ir ⊗ 2 ir r=1 X where ui1 ,...,uir V1 are homogeneous elements with ui1 = 0. By Lemma 2.2, wi1 ,...,wik are End (V )-linearly∈ independent vectors, and there is an6 even element x U(g ) such g2 2 ∈ 2 that xwis = δ1swi1 . Therefore, k ∗ xv = ( 1)|x||uir |u xw +( 1)|x||σ (uir )|σ∗(u ) xσ (w ) − ir ⊗ ir − 1 ir ⊗ 2 ir r=1 X = u w + σ∗(u ) σ (w ). i1 ⊗ i1 1 i1 ⊗ 2 i1 Now, let v1 V1 be any element and j I, then there is a U(g1) (because V1 is simple) and an even∈ element b U(g ) (by Lemma∈ 2.2) such that au∈ = v and bw = w . Hence ∈ 2 i1 1 i1 j ∗ b (a(xv)) = b (aui1 wi1 + aσ1 (ui1 ) σ2(wi1 )) ⊗ ⊗ ∗ =( 1)|b||v1|v bw +( 1)|b||σ1 (v1)|σ∗(v ) bσ (w ) − 1 ⊗ i1 − 1 1 ⊗ 2 i1 = v w + σ∗(v ) σ (w ). 1 ⊗ j 1 1 ⊗ 2 j Since v w +σ∗(v ) σ (w ) v V , j I generates V , we have that U(g g )v = { 1 ⊗ j 1 1 ⊗ 2 j | 1 ∈ 1 ∈ } 1 ⊕ 2 V . Thus V is a simple g1 g2-module. ⊕ Definition 2.5. Using the notation of Proposition 2.4, we define the irreducible tensor product of V1 and V2 as V V , if V V is simple, V ˆ V = 1 ⊗ 2 1 ⊗ 2 1⊗ 2 V, if V V is not simple, ( 1 ⊗ 2 where V is the (unique) simple submodule of V V , obtained in the proof of Proposition 1 ⊗ 2 2.4, for which we have an isomorphism of (g g )-modules V V = V V . 1 ⊕ 2 1 ⊗ 2 ∼ ⊕ Lemma 2.6. Let g be a basic classical Lie superalgebra and V be a finite weight g-module. If there is λ h∗ such that W λ = w W hw = λ(h)w for all h h is nonzero for all submodule W∈ V , then V contains{ a∈ simple| g-module. ∈ } ⊂ Proof. Let W V be a nonzero g-submodule such that W λ has minimal dimension. Set ⊂ λ M = U(g)W λ. For all nonzero element m M, (U(g)m) V is non-zero. Since the ∈ λ ⊂ dimension of W λ is minimal, we see that (U(g)m) = W λ, which implies W λ U(g)m. Therefore, U(g)m = M and M is simple. ⊂ 8 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

Theorem 2.7. Let g1 and g2 be basic classical Lie superalgebras, and S1,S2 be commutative associative unital algebras. If V is a simple finite weight (g1 S1 g2 S2)-module, ˆ ⊗ ⊕ ⊗ then V ∼= V1 V2 where V1 and V2 are simple finite weight modules over g1 S1 and g2 S2, respectively.⊗ Moreover, if End (V ) = k for some i =1, 2, then V = V⊗ V . ⊗ gi⊗Si i ∼ ∼ 1 ⊗ 2 Proof. Let v V (λ,µ) be a nonzero vector of weight (λ,µ) h∗ h∗. We have that ∈ ∈ 1 × 2 h uv =( 1)|u||h1|uh v + [h ,u]v = λ(h )uv for all u U(g S ), h h∗. 1 − 1 1 1 ∈ 2 ⊗ 2 1 ∈ 1 η (λ,η) Then W = U(g2 S2)v V is a finite weight module for g2 S2, because W V . Let N be any nonzero⊗ ⊂ (g S )-submodule of W . Then define⊗ HN to be the subspace⊂ of 2 ⊗ 2 Hom k(N,V ) generated by all homogeneous elements ϕ Hom k(N,V ) such that yϕ(w)= |ϕ||y| ∈ N ( 1) ϕ(yw) for all y g2 S2, w N. Notice that H is a nonzero vector space and − ∈ ⊗ ∈ N it is a module over g1 S1, with action (xϕ)(w)= x(ϕ(w)), for all x g1 S1, ϕ H , and w N. ⊗ ∈ ⊗ ∈ Let M∈ HN be a nonzero (g S )-submodule. The map ⊂ 1 ⊗ 1 Ψ : M N V M,N ⊗ → ϕ w ϕ(w) ⊗ 7→ is a nonzero (g S ) (g S )-module homomorphism. By the simplicity of V , Ψ 1 ⊗ 1 ⊕ 2 ⊗ 1 M,N is surjective. Note that (M N)(α,β) = M α N β , therefore Ψ restricts to a surjection ⊗ ⊗ M,N M λ N µ V (λ,µ). Thus, M λ and N µ are nonzero subspaces, since V (λ,µ) = 0. ⊗ → 6 Since W is a finite weight (g2 S2)-module, it has a simple (g2 S2)-submodule Q W such that Qµ = 0, by Lemma 2.6⊗ . Let w Qµ be a nonzero weight⊗ vector. Then ⊂Q = 6 Q ∈ |u||ϕ| U(g2 S2)w by its simplicity. If ϕ H is homogeneous, then ϕ(uw)=( 1) uϕ(w) ⊗ ∈ Q − for all u U(g2 S2). Thus every element of H is completely defined by its value on w. Using this∈ and the⊗ fact that every weight space of V has finite dimension, one can show Q Q that H is a finite weight (g1 S1)-module. Since for every submodule L H we have that Lλ = 0, then HQ has a simple⊗ finite weight submodule P over g ⊂S , by Lemma 6 1 ⊗ 1 2.6. By Proposition 2.4, the (g1 S1) (g2 S2)-module P Q is either simple (this is k⊗ ⊕ ⊗ k ⊗ the case when Endg1⊗S1 (P ) ∼= or Endg2⊗S2 (Q) ∼= ) or it contains a simple submodule K for which P Q ∼= K K. If P Q is simple, then ΨP,Q is an isomorphism. If P Q = K K⊗, then every⊕ nonzero⊗ proper submodule (resp. quotient) of P Q is ⊗ ∼ ⊕ ⊗ isomorphic to K. In particular, ΨP,Q induces an isomorphism between K and V , and hence V = P ˆ Q. ∼ ⊗

Remark 2.8. We point out that both Lemma 2.6 and Theorem 2.7 still hold for a more general class of Lie superalgebras. Namely, the class of all Lie superalgebras g that admit an abelian subalgebra h g acting (via the adjoint action) semisimple on g. In this case, we would consider the class⊆ of all g-modules on which h acts semisimple. In particular, this is the case for any abelian Lie algebra.

3. Parabolic induction theorem In this section we prove a parabolic induction theorem, which reduces the classification of simple finite-weight modules over to the classification of the so-called cuspidal modules over cuspidal Lie subalgebras of G. G CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 9

Lemma 3.1. Let V be a simple ﬁnite weight -module. Let α ∆ and a A a non zero G ∈ ∈ element. Then either xα a acts locally nilpotently everywhere on V or it acts injectively everywhere on V . ⊗ Proof. Similarly to the Lie algebra case, this result follows from the fact that the set of vectors of V such that x a acts locally nilpotently is a submodule of V , along α ⊗ with the fact that every element xβ with β ∆ acts nilpotently on g via the adjoint representation. ∈

Lemma 3.2. (1) If α is an odd root such that 2α ∆, then xα 1 acts nilpotently on V if and only if x 1 acts nilpotently on V∈. ⊗ 2α ⊗ (2) If α is an odd root such that 2α is not a root, then xα a acts nilpotently everywhere on V . In particular, every odd root acts nilpotently⊗ everywhere on V when g is of the type I. Proof. Both results follow from the fact that 2 2xαv = [xα, xα]v for all α ∆ and v V . ∈ 1 ∈ Proposition 3.3. Let V be a simple finite weight -module. Suppose α ∆0 or α ∆1 with 2α ∆. Then the following conditions are equivalentG ∈ ∈ ∈ (1) For each λ Supp V , V λ+nα is zero for all but finite many n> 0. (2) There is λ ∈Supp V such that V λ+nα is zero for all but finite many n> 0. ∈ (3) For all a A, xα a acts nilpotently on V . (4) x 1 acts∈ nilpotently⊗ on V . α ⊗ Proof. It is clear that (1) implies (2), (3) implies (4). If (2) is true, then xα a acts λ ⊗ nilpotently on V . By Lemma 3.1, xα a acts nilpotently everywhere on V . Now suppose (4) is true. If α ∆ is even, we can use⊗ the same argument of [27, Proposition 2.2] in the ∈ map Lie algebra case. Assume α ∆1 with 2α ∆0. Suppose there is infinitely many n λ+nα ∈ 2 ∈ such that V = 0. Since 2(xα 1) = x2α 1, we have that xα 1 acts nilpotently on V if and only if x6 1 acts nilpotently⊗ on V⊗. However, V λ+2nα =0⊗ or V λ+α+2nα = 0 for 2α ⊗ 6 6 infinitely many n 0, which contradicts the fact x2α 1 acts nilpotently on V (notice that 2α is an even≥ root and we already proved the statement⊗ for this case). Therefore, V λ+nα is zero for all but finite many n> 0. Definition 3.4. Let α ∆. We say that α is locally finite on V if one, hence all, of ∈ conditions of Proposition 3.3 holds. Similarly, we say that α is injective if xα 1 acts injectively on V , and we denote the set of all injective roots on V by inj V . ⊗ We say a subset R ∆ is closed if α + β R whenever α, β R and α + β ∆. For each R ∆ we define⊂ R = α α R .∈ A closed set R ∆∈ is called a parabolic∈ set if R ⊂R = ∆. − {− | ∈ } ⊂ ∪ − Lemma 3.5. If V is a simple finite weight -module, then inj V is closed. G Proof. Let α, β inj V such that α + β ∆. By Lemma 3.1 and Proposition 3.3, xα 1 and x 1 acts∈ injectively everywhere on∈ V . Thus (x 1)(x 1) acts injectively,⊗ so β ⊗ α ⊗ β ⊗ V λ+n(α+β) is non zero for some λ Supp V and all n 0. Applying Lemma 3.1 and Proposition 3.3, x 1 acts injectively∈ on V . ≥ α+β ⊗ 10 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

Following [16], we define the shadow of V as follows. First, we define CV as the saturation of the monoid Z inj V , i.e. + ⊆ Q C = λ mλ Z inj V for some m> 0 . V { ∈Q| ∈ + } i f + − Secondly, we decompose ∆ into four disjoint sets ∆V , ∆V , ∆V , ∆V defined by ∆i = α ∆ α C , V { ∈ | ± ∈ V } ∆f = α ∆ α / C , V { ∈ | ± ∈ V } ∆+ = α ∆ α / C , α C , V { ∈ | ∈ V − ∈ V } ∆− = α ∆ α C , α / C . V { ∈ | ∈ V − ∈ V } Finally, we define the following subspaces of g g+ = gα, gi = h gα, g− = gα. V V ⊕ V + α∈∆i − αM∈∆V MV αM∈∆V − 0 + The triple (gV , gV , gV ) is called the shadow of V . Remark 3.6. Let V be a simple weight -module. G (1) If α, β C , then α + β C . ∈ V ∈ V (2) If α CV , then there is m Z+ such that α := mα Z+inj V satisfies the following∈ condition: for any weight∈ λ Supp V , we have that∈ V λ+nα = 0 for all ∈ e 6 { } n 0. e (3) The≥ following observation is an important difference between super and non su- i per setting: the set ∆V is not necessarily equal to inj V . Indeed, since the Lie superalgebra g = D(2, 1, a) admits cuspidal modules [16], we can choose V such i that ∆V = ∆. Notice however that there is no odd root in inj V , as all roots in ∆1 are locally finite (2α / ∆ for any α ∆1). In this case, we actually have that ∆ inj V , and 2( ε ∈ε ε )=( 2∈ε )+( 2ε )+( 2ε ) Z ∆ C . 0 ⊆ ± 1 ± 2 ± 3 ± 1 ± 2 ± 3 ∈ + 0 ⊆ V Let V be a simple weight -module. For α ∆ and λ Supp V , we define the α-string through λ to be the set x G Q λ + α Supp∈ V . The∈ following lemma describes the shadow of V in terms of{ strings.∈ | ∈ } f Lemma 3.7. (1) α ∆V if and only if the α-string through any λ Supp V is bounded. ∈ ∈ i (2) α ∆V if and only if the α-string through any λ Supp V is unbounded in both directions.∈ ∈ + (3) α ∆V if and only if the α-string through any λ Supp V is bounded from above only.∈ ∈ − (4) α ∆V if and only if the α-string through any λ Supp V is bounded from below only.∈ ∈

+ i Let V be a simple ﬁnite weight -module. Then, it follows from Lemma 3.5 that gV , gV , − G + + i i − − and gV are subalgebras of g. In particular, V = gV A, V = gV A, and V = gV A are subalgebras of . G ⊗ G ⊗ G ⊗ G Proposition 3.8. Let V be a simple ﬁnite weight -module. G CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 11

f f (1) If ∆0 ∆V , then V is finite-dimensional. In this case, ∆ = ∆V , and V is a simple⊂ highest weight module. (2) If ∆0 inj V , then V is bounded. Furthermore, there is a finite subset Θ Supp V ⊂ µ λ ⊂ such that Supp V = Θ+ 0, and dim V = dim V if there is γ Θ such that µ,λ γ + . Q ∈ ∈ Q0 f Proof. (1): Suppose ∆ = ∆ . Write ∆0 = α1,...,αt and ∆1 = β1,...,βs . Let λ Supp V , and define { } { } ∈ W (λ)= (g A) V λ = (g A/Ann (V )) V λ, 0 1 ⊗ 1 ⊗ A W (λ)=Û(gαi A) ...U(^gα1 A)U(h A)W (λ), i ⊗ ⊗ ⊗ 0 where i = 1,...,t. Since W (λ) is a weight h 1-module, we can define S (λ) as its i ⊗ i weights. By the PBW theorem and the simplicity of V , Wt(λ)= V and St(λ) = Supp V . W0(λ) has finite dimension and S0(λ) is finite, because A/Ann A(V ) and, therefore, λ (g1 A/Ann A(V )) has finite dimension and V has finite dimension. Therefore, by Proposition⊗ 3.3, V S (λ)= Supp V γ + nα i =1,...,s, n 0 1 ∩{ i | ≥ } γ∈[S0(λ) is a finite set. Suppose, by induction, that Si(λ) is finite. By Proposition 3.3, for each γ S (λ) the set Supp V γ + nα n 0 is finite. Therefore, S = ∈ i ∩ { i | ≥ } i+1 Supp V γ + nαi n 0 is finite union of finite sets, we see Si+1(λ) is a fi- γ∈Si(λ) ∩{ | ≥ } nite set as well. Consequently, Supp V is a finite set. Since V has a finite number of Sweights and it has finite-dimensional weight spaces, V is a finite-dimensional -module. (2): Suppose ∆ inj V . As we did in the first part, we define G 0 ⊂ W (λ)= (g A) V λ = (g A/Ann (V )) V λ, 0 1 ⊗ 1 ⊗ A for some fixed λ Supp V .^ By the same argument^ given in the first part, W0(λ) has finite dimension, thus the∈ set Θ of its weights is finite. By the PBW Theorem and the simplicity α of V , we have that V = U(g A)U(h A)W0(λ), thus Supp V =Θ+ 0 since α∈∆0 ⊗ ⊗ Q ∆0 inj V . ⊂ Q γ+nα Let α ∆0 and γ Θ, then α, α inj V and V = 0 for all n Z. Hence V has infinite dimension.∈ Since∈ x 1 acts− injectively,∈ the linear6 map ∈ α ⊗ V γ V γ+α → v (x 1)v 7→ α ⊗ γ γ+α γ+α is injective, therefore dim V dim V . Likewise, x−α 1 acts injectively on V and thus the linear map ≤ ⊗ V γ+α V γ → v (x 1)v 7→ −α ⊗ is injective, which implies that dim V γ+α dim V γ. Repeating this argument we conclude γ γ+β ≤ that dim V = dim V for all n Z and β 0. Since Θ is finite and dim V γ is finite∈ for all∈γ Q Θ, we conclude V is bounded. ∈ 12 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

Corollary 3.9. Assume that V is a simple infinite-dimensional weight -module. Then inj V ∆ = . G ∩ 0 6 ∅ Definition 3.10 (Triangular decomposition). A triangular decomposition T of g is a + 0 − decomposition g = gT gT gT such that there exists a linear map l : Z for which + α 0 ⊕ ⊕ α − α Q→ gT = l(α)>0 g , gT = l(α)=0 g and gT = l(α)<0 g . A triangular decomposition T + 0 − • • of is a decomposition of the form = T T T , where T := gT A and T is a G L L G G ⊕L G ⊕ G G 0 ⊗ triangular decomposition of g. A triangular decomposition is proper if gT = g. Finally, we − 6 set ∆+ := α ∆ l(α) > 0 , ∆ := α ∆ l(α) < 0 and ∆0 := α ∆ l(α)=0 . T { ∈ | } T { ∈ | } T { ∈ | } Lemma 3.11. Let V be a simple finite weight -module. i G i (1) The monoid V generated by all even roots in ∆V is a group, and for every odd i Q i root α ∆V there is m> 0 such that mα V . ∈ ∈ Q i 0 + + (2) There is a triangular decomposition T of such that ∆V = ∆T , ∆V ∆T , and ∆− ∆−. G ⊂ V ⊂ T i Proof. Since the root system of coincides with that of g and the set ∆V is completed determined by the action of g onGV (see Proposition 3.3 and Lemma 3.1), the same proof of [16, Theorem 3.6] also work in our setting. 0 Let T be a triangular decomposition of . For any weight T -module W , we define the induced module G G MT (W )= U ( ) 0 + W, G ⊗U(GT ⊕GT ) where the action of + on W is given by +W = 0. GT GT 0 0 Proposition 3.12. Let W be a weight T = gT A-module whose support is included in a single T -coset, where T is the rootG lattice of⊗g0 . Q Q T (1) MT (W ) has a unique submodule NT (W ) which is maximal among all submodules of MT (W ) with trivial intersection with W . (2) NT (W ) is maximal if and only if W is simple. In particular, LT (W )= MT (W )/NT (W ) 0 is a simple -module if and only if W is a simple T -module. (3) If W is simple,G the space G G+ + L (W ) T = v L (W ) xv =0 for all x T ∈ T | ∈ GT + of T -invariants is equal to W . G Proof. This proof is standard (see [16, Lemma 2.3, Corollary 2.4]). Theorem 3.13. Let V be a simple finite weight -module, then there is a triangular − 0 + i G 0 + + − − decomposition ∆ = ∆T ∆T ∆T such that ∆V = ∆T , ∆V ∆T , ∆V ∆T , the vector + ∪ + ∪ ⊂ ⊂ + space of -invariants V GT is a simple bounded i -module, and V = L (V GT ). GT GV ∼ T − 0 + Proof. By Lemma 3.11, there is a triangular decomposition ∆ = ∆T ∆T ∆T such that i 0 + + − − 0 i ± ± ∪ ∪ i + + ∆V = ∆T , ∆V ∆T , and ∆V ∆T , i.e. T = V , and V T . Since [ V , T ] T , + ⊂ ⊂ G G G ⊂+ G G G ⊂ G GT i GT i V is a submodule of V over V . First, we will prove that V is a nonzero V -module, Gi G and we will use it to create a V -module W such that its support is included in a single T G G+ -coset. Then, we will prove that V ∼= LT (W ), W is simple, and W = V T . Lastly, we Qwill show that W is bounded. CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 13

+ Let v V be a nonzero weight vector of weight λ. Using the fact all roots ∆V are ∈ + locally finite, and the argument given Proposition 3.8 (1), U( T )v is a finite dimensional + + G + + + GT GT i GT T -module. Therefore, U( T )v V = 0, and V is a nonzero V -module. Let w V G G ∩ 6 i G i ∈ any weight nonzero vector and let W = U ( V ) w. Thus, W is a nonzero weight V -module whose support is included in a single T -coset.G G The linear map Q ϕ : M (W ) V T → u i + v uv ⊗U(GV ⊕GT ) 7→ is a well-defined -module homomorphism. Furthermore, ϕ is surjective, because its image contains WGand V is a simple -module. Hence V is isomorphic to M (W )/ker ϕ, G T and ker ϕ is a maximal submodule of MT (W ). By the PBW theorem, MT (W ) can be − − identified with U( T ) T W 1 W thus ϕ(1 v)= v for all v W , and ϕ restricted i G G ⊗ ⊕ ⊗ ⊗ ∈ to W is a V -module homomorphism and an injection that preservers weight spaces. Therefore, WG ker ϕ = 0, and ker ϕ N (W ). By Proposition 3.12 (1), ker ϕ = N (W ), ∩ ⊂ T T because ker ϕ is maximal. We conclude that V = LT (W ). By Proposition 3.12, W is + ∼ simple and it is equal to V GT . i Since gV is a good Levi subalgebra, we have a isomorphism gi = z l, V ∼ ⊕ z h g l k lr where is a subalgebra of the Cartan subalgebra of , and ∼= r=1 is a direct sum of certain simple finite-dimensional Lie superalgebras where at most one li has nontrivial odd part (see [16, 17] for a complete list of good Levi subalgebras).L Since z is a subalgebra i of h, W is simple as a gV A-module if and only if W is simple as a l A-module. Since at most one li has nontrivial⊗ odd part, Theorem 2.7 implies that ⊗ k W = W , |l⊗A ∼ r r=1 O r where Wr is a simple finite weight module over l A for each r =1,...,k. By definition, i ⊗ r all even roots on ∆V are injective. By Proposition 3.8, each Wr is a bounded l A- i ⊗ r module. Therefore, W is bounded gV A-module, due to the orthogonality between l and ls. ⊗ Definition 3.14 (Cuspidal modules). Let V be a simple weight -module. When there is a proper triangular decomposition T of and a simple 0 -moduleG W such that V = G GT ∼ LT (W ), we say V is parabolic induced. If V is not parabolic induced, we call V cuspidal. If admits cuspidal modules we say is a cuspidal superalgebra. G G By Theorem 3.13, every finite weight -module is either cuspidal, or it is isomorphic to G 0 some parabolic induced module of the form LT (W ) with W being a bounded T -module. However, it is not the case that every parabolic induced module constructedG in this way is a finite weight -module, as can be seen in the following example. G Example 3.15. Let g be a basic classical Lie superalgebra and A = k[t]. Any usual triangular decomposition ∆ = ∆+ ∆− defines a triangular decomposition T of ∆ in the sense of Definition 3.10. In particular,∪ we have 0 = h k[t]. Fix a nonzero element GT ⊗ 14 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

h h, and let Λ : h k[t] k be a linear functional satisfying Λ(h tk)= k for all k 0. The∈ 1-dimensional ⊗0 -module→ kv defined by xv = Λ(x)v is a simple⊗ 0 -module with≥ GT Λ Λ Λ GT finite-dimensional weight-spaces, but the simple highest weight -module LT (kvΛ) does not have finite-dimensional weight spaces by [30, Theorem 4.16].G

+ i GT Remark 3.16. Assume that the gV A-module V is cuspidal. By Proposition 3.13, + ⊗ GT i we have that V is bounded, and hence it is bounded when restricted to gV . Thus, it + i i GT has ﬁnite length over gV by [16, Lemma 6.2]. If M is a simple gV -submodule of V , then M is cuspidal. In particular, gi = z l is a cuspidal Levi superalgebra, with z h V ∼ ⊕ ⊂ and l isomorphic to osp(1 2), osp(1 2) sl2, osp(n 2n) with 2 < n 6, D(2, 1; a), or a reductive Lie algebra with| irreducible| components⊕ of| type A and C (see≤ [16, 17]).

4. Cuspidal Bounded Modules In this section we complete the classiﬁcation given in Theorem 3.13. By Remark 3.16, we just need to classify all cuspidal bounded simple modules over g A, where g is ⊗ isomorphic to osp(1 2), osp(1 2) sl2, osp(n 2n) with 2 < n 6, D(2, 1; a), or to a reductive Lie algebra| with irreducible| ⊕ components| of type A and C≤. Since the case where g is isomorphic to a Lie algebra is known [4], it only remains to classify all cuspidal bounded simple modules over g A, for g isomorphic to osp(1 2), osp(n 2n) with 2 < n 6, or D(2, 1; a). ⊗ | | ≤

Proposition 4.1. Suppose that g is a basic classical Lie superalgebra such that g0 is semisimple. If V is a simple bounded weight g A-module, then the ideal Ann A(V ) is radical. ⊗

Proof. Set I := Ann A(V ). Then A/I is a finite-dimensional commutative algebra and V is a faithful representation of g A/I. Since A/I is artinian, its nilradical J = √I/I ⊗ is nilpotent. If J = 0, then I = √I is radical and the proof is complete. Seeking a contradiction, let’s assume that J = 0. Then there is m > 0 such that J m−1 = 0, but ′ J m = 0. Let m′ be the smallest positive6 integer greater or equal to m/2, and set N6 = J m . Thus N = 0, and N 2 = 0. Claim6 1: gα N acts nilpotently on V for every α ∆. If α ∆ , x ⊗gα, and a N, then ∈ ∈ 1 ∈ ∈ 0 = ([x, x] a2)u = [x a, x a]u = 2(x a)2u, ⊗ ⊗ ⊗ ⊗ for all u V , because a2 N 2 = 0. Therefore, (x a)2u = 0, and gα N acts nilpotently on V for∈ every odd root α∈ ∆. If α is even, the proof⊗ that gα N acts⊗ nilpotently on V follows from Step 2 of [4, Proposition∈ 4.4]. ⊗ Claim 2: There is a nonzero weight vector w V such that (g N)w =0. Let λ Supp V . Note that h N is an abelian∈ Lie algebra with⊗ finite dimension that commutes∈ with h 1, thus (h ⊗N) V λ V λ, and, by Lie’s Theorem, there is a nonzero λ ⊗ ⊗ ⊂ vector v0 V such that (h N)v0 kv0. ∈ + − ⊗ ⊂ ± α Let ∆ = ∆ ∆ be a choice positive roots of ∆ and set n := α∈∆± g . Recall that N is an ideal of∪A/I, N = 0, and N 2 = 0, thus [g N, g N] [g, g] N 2 = 0. By Claim 1, every element of the finite-dimensional6 abelian Lie⊗ algebra⊗ n⊂+ NL⊗acts nilpotently on V . Thus U(n+ N)v is a finite-dimensional (n+ N)-module.⊗ By Engel’s theorem, there ⊗ 0 ⊗ CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 15

+ + is a nonzero vector u U(n N)v0 such that (n N)u = 0. Since U(n+ N)v0 V is a weight module, we∈ can assume⊗ that u is a weight⊗ vector. The finite-dimensional⊗ ⊂ abelian Lie algebra n− N acts nilpotently on u, and a similar argument shows that there is a nonzero weight⊗ vector w U(n− N)u such that (n− N)w = 0. Since U(n+ N) commutes with U(n− N∈) (due to⊗ N 2 = 0), we have (n+⊗ N)w = 0. Since ⊗ − + ⊗ ⊗ − + (h N)v0 kv0, w U((n n ) N)v0, and h N commutes with U((n n ) N), we⊗ conclude⊂ that w ∈is an eigenvector⊕ ⊗ for every element⊗ of h N. ⊕ ⊗ ⊗ It remains to show that (h N)w = 0. The proof that (hα t)w = 0 for all t N and α ∆ is the same as that⊗ of Step 5 of [4, Proposition 4.4].⊗ Since we are assuming∈ ∈ 0 g0 is a semisimple Lie algebra and h is a Cartan subalgebra of g0 (by definition), the set hα α ∆0 generates h as a vector space. Hence (h N)w = 0 as we wanted. { Claim| ∈ 3: }The ideal I is radical. ⊗ Let W = w V (g N)w =0 . Since N is an ideal, for all (x a) g A/I and y b g N{ we∈ get| that⊗ } ⊗ ∈ ⊗ ⊗ ∈ ⊗ (y b)(x a)w =( 1)|x||y|(x a)(y b)w + ([x, y] ba)w =0. ⊗ ⊗ − ⊗ ⊗ ⊗ Together with Claim 2, we obtain that W is a nonzero g A/I-submodule of V . Since V is simple, W = V , and hence 0 = N Ann (V ), which⊗ contradicts the fact that V is 6 ⊂ A/I a faithful representation of g A/I (since (g N)V = 0). Hence J = √I/I = 0, which ⊗ ⊗ implies that I = Ann A(V ) is a radical ideal of A. Definition 4.2 (Evaluation modules). Suppose m ,..., m Specm A are pairwise dis- 1 r ∈ tinct. The associated evaluation map evm1,...,mr is the composition r r r evm m : / g m = (g A/m ) = g . 1,..., r G → G ⊗ i ∼ ⊗ i ∼ i=1 ! i=1 Y M If Vi is a g-module with respective representations ρi : g End Vi for each i = 1,...,r, then the composition → r r evm ,...,m N ρ 1 r gr i=1 i End V G −−−−−→ −−−−→ i i=1 ! is a representation of called an evaluation representationO. The corresponding module is called an evaluation moduleG , and is denoted by r mi Vi . i=1 O For each simple finite weight -module (or g-module) V , we denote R = C ∆. G V V ∩ Notice that if V is cuspidal, then RV is equal to ∆.

Lemma 4.3. Let r 2, and V1,...,Vr be simple bounded weight g-modules such that r ≥ r ∆= i=1RVi . Then the g-module V := i=1 Vi is bounded if and only if dim Vi = for at most∪ one i =1,...,r. ∞ N Proof. It is clear that if dim V = for at most one i =1,...,r, then V is bounded. i ∞ Assume without loss of generality that dim V1 = dim V2 = . Let’s prove that V is not ∞1 α bounded. Notice that the fact that RV1 is closed implies that g := hα g α∈RV1 ⊕ α∈RV1 is a subalgebra of g. L L 16 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

r r α If (RV1 + i=2RVi ) ∆= or (RV1 + i=2RVi ) ∆ RV1 , then hα g ∪ ∩ ∅ ∪ ∩ ⊆ α∈RV1 ⊕ α∈RV1 is an ideal of g, which gives g = g1, since g is simple. Thus, ∆ = R and R R . L V1 LV2 ⊆ V1 For α RV2 (which is nonempty, as dim V2 = ), pick m Z+ so that α = mα (Z inj V∈ ) (Z inj V ). Now, for any r-tuple of∞ weights (λ ,...,λ∈ ) r Supp V , we∈ + 1 ∩ + 2 1 r ∈ ×i=1 i have e λ1+···+λr+nα lim (dim V e)= n→∞ ∞ as, for each n 0, the following inclusion holds ≥ n λ1+lα λ2+kα λ3 λr λ1+···+λr+nα V e V e V V V e. 1 ⊗ 2 ⊗ 3 ⊗···⊗ r ⊆ l+Mk=n Assume now that there are α R and β R for j 2 such that α + β r R . ∈ V1 ∈ Vj ≥ ∈ ∪i=2 Vi Suppose without loss of generality that j = 2 and ﬁx any r-tuple of weights (λ1,...,λr) r ∈ i=1Supp Vi. We claim that there is m Z+ such that α := mα and β := mβ satisfy the ×following condition ∈ λ1+···+λr+n(α+β) lim dim(V e e )=e e n→∞ ∞ Indeed, for all n 0, one of the following hold: (1): α + β ≥R . In this case, we take m Z for which α = mα Z inj V , ∈ V2 ∈ + ∈ + 1 β = mβ Z inj V and m(α + β)= α + β Z inj V . Then ∈ + 2 ∈ + 2 n e λ1+lα λ2+lβ+(n−l)(α+β) λ3 λr λ1+···+λr+n(α+β) e V e V e e e e V V V e e 1 ⊗ 2 e ⊗ 3 ⊗···⊗ r ⊆ Ml=0 (2): α + β RVk for k = 3,...,r, and (for convenience of notation, assume that k = 3). Similarly∈ to the previous case, take m Z such that α = mα Z inj V , ∈ + ∈ + 1 β = mβ Z inj V and m(α + β)= α + β Z inj V . Then ∈ + 2 ∈ + 3 n e λ1+lα λ2+lβ λ3+(n−l)(α+β) λ4 λr λ1+···+λr+n(α+β) e V e V e V e ee V V V e e 1 ⊗ 2 ⊗ 3 e ⊗ 4 ⊗···⊗ r ⊆ Ml=0 In any case, we conclude that lim (dim V λ1+···+λr+n(α+βe))= . n→∞ e ∞ Theorem 4.4. Suppose g0 is semisimple (in particular, this is the case when admits cuspidal modules). If V is a simple cuspidal bounded -module, then V is isomorphicG to an evaluation module. G

Proof. By Proposition 4.1, I = Ann A(V ) is radical, so there are distinct maximal ideals m m r m 1,..., r of A such that I = i=1 i. By the Chinese Remainder Theorem, r r T / g m = g (A/m A/m ) = (g A/m ) = g⊕r, G ⊗ i ∼ ⊗ 1 ⊕···⊕ 1 ∼ ⊗ i ∼ i=1 ! i=1 \ M where the last isomorphism follows from the fact that A is a finitely generated algebra over the algebraically closed field k. In particular, V is a simple module over g⊕r. By ˆ ˆ Theorem 2.7, V ∼= V1 ... Vr, where V1,...,Vr are simple finite weight g-modules. Now, notice that the⊗ g-module⊗ V ˆ ... ˆ V is bounded if and only if the g-module 1⊗ ⊗ r V := V V is bounded. Moreover, since V is cuspidal, we have that ∆ = R . 1 ⊗···⊗ r V e CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 17

Since either V = V V or V = V (see Proposition 2.4) we conclude that R = r R . ∼ ⊕ ∼ V i=1 Vi Indeed, this follows from Lemma 3.7 along with the fact that Supp V = Supp V Supp V , S λ e λei × and V = λ1+···+λr=λ Vi . Thus, by Lemma 4.3, at most one of the Vi has inﬁnite dimension. Without loss of generality, we assume that dim V1 = e, and dim Vi < for N ∞ ∞ each ie=2,...,r. In particular, for all i> 1, the g-modules Vi are highest weight modules, which implies that Endg Vi ∼= k. Now Theorem 2.7 gives us that V = V ˆ ... ˆ V = V V . ∼ 1⊗ ⊗ r 1 ⊗···⊗ r In terms of representations, we obtained that the representation ρ : End(V ) associated to the -module V factors through the following composition G → G r r evm ,...,m ⊗ ρ g (A/Ann (V )) 1 r g⊕r i=1 i End V , G → ⊗ A −−−−−→ −−−−→ i i=1 ! O where ρi : g End(Vi) is the representation associated to the g-module Vi for each i = 1,...,r. But→ this composition is precisely the representation associated to the evaluation m r i module i=1 VI . The proof is completed.

Proposition 4.5. Let V1,...,Vr be simple finite weight g-modules and m1,..., mr pairwise N m distinct maximal ideals of A. Then the evaluation -module V := r V i is a cuspidal G i=1 i bounded module only if dim Vi = for precisely one Vi, in which case Vi is a cuspidal bounded g-module. In particular, if∞V is a cuspidal bounded -module,N then it is simple. G Proof. If V1,...,Vr are bounded and no more than one of them is infinite-dimensional, then it is clear that V is bounded. On the other hand, if V is bounded and N > 0 is such that dim V λ N for each λ Supp V , then the dimension of the weight spaces of each ≤ ∈ Vi has to be less or equal to N as well. Thus each Vi is bounded, and, by Lemma 4.3, no more than one Vi can be infinite-dimensional. This proves the first statement. k The second statement follows from Theorem 2.7 along with the fact that Endg(Vi) ∼= for all, except at most one i =1,...,r.

5. Affine Kac-Moody Lie superalgebras 5.1. Finite weight modules over aﬃne Lie superalgebras. The aﬃne Lie superalgebra (g) associated to a basic classical Lie superalgebra g is the universal central extensionA of the loop superalgebra L(g) := g C[t, t−1]. Concretely, if ( , ) denotes an invariant supersymmetric non degenerate even⊗ bilinear form on g, then (·g)· is the vector space L(g) Cc for an even element c, and the bracket is given by A ⊕ [c, (g)]=0, [x tn,y tm] = [x, y] tm+n + n(x, y)δ c, A ⊗ ⊗ ⊗ n,−m for any x tn, y tm L(g). Notice that we have a short exact sequence of Lie superalgebras⊗ ⊗ ∈ 0 Cc (g) L(g) 0. → →A → → Recall that (h) := h Cc is a Cartan subalgebra of (g). It is clear that any (g)-module VAon which c⊕acts trivially is naturally a moduleA over the Lie superalgebra A(g)/Cc = L(g). On the other direction, any L(g)-module V can maid into a (g)- A ∼ A module with trivial action of c via the. Let Vc=0 denote such (g)-module. The main result of this section is the following: A 18 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

Theorem 5.1. The assignment that maps an L(g)-module V to the (g)-module Vc=0 provides a bijection between the set of simple finite weight L(g)-modulesA and the set of simple finite weight (g)-modules. Moreover, this bijection restricts to a bijection between the respective sets simpleL cuspidal modules. Proof. The fist statement will follow if we show that c acts trivially on any simple finite weight (g). To prove this, we notice that c must act as a scalar k on any simple (g)- module,A by Lemma 2.1. We claim that k is zero. Indeed, let h h be a nonzero elementA for which (h, h) = 0. Let V λ any weight space. Since dim V λ <∈ , we can take the trace −16 λ ∞ of [h t, h t ] EndK(V ), which is zero. On the other hand, the equality ⊗ ⊗ ∈ [h t, h t−1]=(h, h)c ⊗ ⊗ shows that this trace is precisely dim V λ (h, h)k. The first statement follows. The fact that c acts trivially on any simple finite weight (g)-module implies the second statement. A 5.2. Loop modules over affine Kac-Moody Lie superalgebras. Recall that the affine Kac-Moody Lie superalgebra g associated to a basic classical Lie superalgebra g is a one-dimensional extension of (g) by a derivation d such that [d, x tk] = kx tk. A ⊗ ⊗ Thus, g = (g) Cd and the Cartanb subalgebra of g is given by h = h Cc Cd. It is well knownA that⊕ any simple finite weight module of g of nonzero level is parabolic⊕ ⊕ induced from ab cuspidal simple module over some cuspidal subalgebrab of g b[19]. In particular, if g admits cuspidal modules, then all such modules mustb be of level zero. In this subsection, we show that g admits cuspidal modules if and only of L(g) does, by providing a familyb of cuspidal g-modules constructed from cuspidal L(g)-modules. For any L(gb)-module V we can consider the the loop g-module associated to V to be the vector space L(V )= V C[t, t−1] where b ⊗ (x tr)(v ts)=(x tr)v tr+s, cL(V )=0,b d(v tr)= rv tr. ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ It is clear that the L(g)-module L(V ) is a weight module with infinite-dimensional weight spaces. However, as a module over h we have that L(V ) = V λ C[t, t−1] = V λ tr, ∼ ⊗ b ∼ ⊗ Z λ∈MSupp V λ∈SuppMV, r∈ and hence Supp L(V ) = λ + rδ λ Supp V, r Z and dim L(V )λ+rδ = dim V λ for all λ Supp V , r Z. Thus,{ L(|V )∈ is a finite weight∈ } (bounded) module if and only if V is.∈ Since from now∈ on our goal is to study g-modules of the form L(V ), we lose no information by assuming that g = L(g) Cd. ⊕ λi Let V0,...,Vk be cyclic weight g-modules generatedb by weight vectors vλi Vi for × ∈ i = 0,...,k. For a0,...,ak bC such that ai = aj if i = j, define a homomorphism of algebras χ : U(L(h) Cd∈) C[t, t−1] by extending6 6 (λ,a) ⊕ → k χ (h tn)= anλ (h) tn, χ (d)=0. (λ,a) ⊗ i i (λ,a) i=0 ! X It follows by [7, Section 4] that im χ = C[tr, t−r] for some r 0. For any n Z, set (λ,a) ≥ ∈ v = k v tn and let v = v . λ,n i=0 λi ⊗ λ λ,0 N CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 19

Let g = n− h n+ be a triangular decomposition of g. In what follows, for λ h∗, we let F (λ) denote⊕ the⊕ simple highest weight g-module of highest weight λ. The following∈ result is a generalizations or results from [14, Sections 1 and 4]. Theorem 5.2. With the above notation, the following statements hold: (1) For any a C× and any g-module V , we have that L(V a) is simple if and only if V is simple∈ (2) Assume that the map χ is surjective. Then L(V a0 V ak ) is cyclic, (λ,a) 0 ⊗···⊗ k generated by any vector of the form vλ,n for n Z. ∗ ∈ (3) Let λ1,...,λk h and consider F (λ1),...,F (λk) the respective simple highest weight g-modules.∈ Let V be a simple weight g-module and a C× such that 0 0 ∈ a0 = ai for i = 1,...,k. If there is λ0 Supp V0 for which χ(λ,a) is surjective, 6 a0 ∈ then g-module L(V F (λ )a1 F (λ )ak ) is simple. 0 ⊗ 1 ⊗···⊗ k Proof. (1): If W V is a proper submodule then L(W ) is a proper submodule of L(V ). Conversely, ifbV is⊆ simple, then the statement follows from the fact that V tn = U(g)v tn for any vector v V , n Z. ⊗ ⊗ (2): Set N = ∈ C∈v . Using that a = a for i = j, we can prove that m∈Z λ,m i 6 j 6 u v u v tm U(L(g))N P 0 λ0 ⊗···⊗ k λk ⊗ ∈ for any u0,...,xk U(g). Moreover, the fact that χ(λ,a) is surjective, implies that we can ′ ∈ ′ −1 ﬁnd H,H U(L(h)) such that χ(λ,a)(H)= t and χ(λ,a)(H )= t . In particular, for any ∈ p ′ p ﬁxed n Z and any m Z, we have that either vλ,m = H vλ,n or vλ,m = (H ) vλ,n for some p ∈ N. Then N ∈U(L(h))v and L(V a0 V ak ) is cyclic, generated by any ∈ ⊆ λ,n 0 ⊗···⊗ k vector of the form vλ,n for n Z. ∈ a0 a1 (3): Fist notice that any weight of L(V0 F (λ1) ) is of the form γ + mδ for some γ Supp V F (λ ), m Z. Moreover, ⊗ ∈ 0 ⊗ 1 ∈

L(V a0 F (λ )a1 )γ+mδ = V µ0 F (λ )λ1−η1 tm 0 ⊗ 1  0 ⊗ 1  ⊗ µ0+(λO1−η1)=γ a0 a1   To prove that L(V0 F (λ1) ) is simple, we ﬁrst prove the following claim: for any ⊗ a0 a1 γ+mδ nonzero weight vector w L(V0 F (λ1) ) , there is u U(g) such that uw V µ0 F (λ )λ1 tℓ for some∈ µ Supp⊗ V and some ℓ Z. ∈ ∈ 0 ⊗ 1 ⊗ 0 ∈ 0 ∈ Write b w = v w tm i ⊗ i ⊗ i ! X for linearly independent weight vectors v V and nonzero weight vectors w F (λ ). i ∈ 0 i ∈ 1 If all wi are highest weight vectors, there is no claim to prove. Suppose then that there is some w which is not a highest weight vector. Notice that for any x n+, n Z we have i ∈ ∈ (x tn)w = anxv w +( 1)|x||vi|anv xw tm+n U(g)w. ⊗ 0 i ⊗ i − 1 i ⊗ i ⊗ ∈ i X |x||vi| m+n Since a0 = a1, we can prove thatw ˜x,n := i( 1) vi xwi t bU(g)w. Notice 6 + − ⊗ ⊗ ∈ + now that there must exist x n , such thatw ˜x,n = 0. Indeed, ifw ˜x,n = 0 for all x n , ∈ P 6 ∈ then since all vi are linearly independent, we must have xwi = 0 for all i. Butb this implies 20 L.CALIXTO,V.FUTORNY,ANDH.ROCHA

that all wi are highest weight vectors, contradicting our assumption. Thusw ˜x,n U(g)w is nonzero for some x n+. Applying this argument recursively, we show that∈ all w ∈ i occurring in some nonzero vector uw U(g)w are highest weight vectors, and henceb v v tℓ U(g)w for some µ Supp∈ V . This proves the claim. µ0 ⊗ λ1 ⊗ ∈ 0 ∈ 0 Finally, by the same argument used in partb (b), we prove that for any u0 U(g) the ℓ ℓ ∈ vector u0vµ0 vλ1 b t lies in U(g)(vµ0 vλ1 t ) U(g)w. Since there is λ0 Supp V0 ⊗ ⊗ ⊗ ⊗ ℓ ⊆ ∈ such that χ(λ,a) is surjective, and vλ0 vλ1 t U(g)w, it follows from part (b) that a0 a1 ⊗ℓ ⊗ ∈ L(V0 F (λ1) )= U(g)(vλ0 vbλ1 t ) U(g)w. Theb proof is complete. ⊗ ⊗ ⊗ ⊆ b r −r We now consider theb case where χ(λ,a) is notb surjective, and hence im χ(λ,a) = C[t , t ] for some r > 1, by [7, Section 4]. Theorem 5.3. With the above notation, the following statements hold: (1) The module L(V a0 V ak ) is generated by the vectors v for i =1,...,r 1. 0 ⊗···⊗ k λ,i − (2) L(V a0 V ak )= r−1 Li(V a0 V ak ), where Li(V a0 V ak ) is the 0 ⊗···⊗ k i=0 0 ⊗···⊗ k 0 ⊗···⊗ k cyclic module generated by vλ,i for i =1,...,r 1. ∗ L − (3) Let λ1,...,λk h and consider F (λ1),...,F (λk) the respective simple highest ∈ × weight g-modules. Let V0 be a simple weight g-module and a0 C such that ∈ r −r a0 = ai for i = 1,...,k. If there is λ0 Supp V0 for which im χ(λ,a) = C[t , t ], 6 a0 a1 ak ∈ r−1 i a0 a1 ak then L(V0 F (λ1) F (λk) )= i=0 L (V0 F (λ1) F (λk) ) i⊗ a0 ⊗···⊗a1 ak ⊗ ⊗···⊗ and each L (V F (λ1) F (λk) ) is simple. 0 ⊗ ⊗···⊗ L Proof. The proof follows the same steps as that of Theorem 5.2, but differently from that ′ r ′ −r case, we now can find H,H U(L(h)) such that χ(λ,a)(H) = t and χ(λ,a)(H ) = t . This implies that, for any m ∈ Z such that m i mod (r), we have v = Hqv , ∈ ≥0 ≡ λ,m λ,i v =(H′)qv , and hence that N r−1 U(L(h))v . λ,−m λ,r−i ⊆ i=0 λ,i Remark 5.4. It follows from Section 4 thatP L(g) admits cuspidal modules if and only if g is isomorphic to osp(1 2), osp(1 2) sl2, osp(n 2n) with 2 < n 6, D(2, 1; a), or to a reductive Lie algebra| with irreducible| ⊕ components| of type A ≤and C. Recall from Theorem 4.4 that any simple bounded cuspidal L(g)-module is of the form V a0 0 ⊗ F (λ )a1 F (λ )ak for some simple bounded cuspidal g-module V , some simple finite- 1 ⊗···⊗ k 0 dimensional g-modules F (λ1),...,F (λk), and some pairwise distinct complex numbers × a0,...,ak C . Therefore, Theorem 5.3 (3) provides a family of simple bounded cuspidal modules over∈ g. In the case of Lie algebras, these are exactly the simple modules that appear in the work of Dimitrov and Grantcharov [15]. Moreover, it is claimed in that paper that theyb exhausts all simple cuspidal bounded g-modules. We end this subsection with the following conjecture: b Conjecture 5.5. If M is a simple bounded cuspidal g-module, then there is a simple bounded cuspidal L(g)-module V for which M = Li(V ) for some i N. ∼ ∈ 5.3. Bounded modules over affine Kac-Moody Lieb superalgebras. In this subsection we assume that g is a basic classical Lie superalgebra of type I in order to define analogs of Kac g-modules in the affine setting. Consider the Z-grading g = g−1 g0 g1, and recall that g0 is a reductive Lie algebra with 1-dimensional center, that g⊕ are⊕ simple g -modules, and that [g , g ] = 0 . b ±1 0 ±1 ±1 { } CLASSIFICATION OF SIMPLE HARISH-CHANDRAMODULES 21

Following [11], for a simple weight g0-module S we deﬁne the Kac module associated to S to be the induced g-module

K(Sb) := U(g) U(g0⊕g1) S, b ⊗ b b where we are assuming that g1S = 0. It is easy to prove that K(S) admits a unique simple quotient which we denote by V (S).b This defines the Kac induction functor from the category of g0-modules tob the category of g-modules, and we have the following main result of this section. Theorem 5.6. bLet g be a basic classical Lie superalgebrasb of type I. Then the functor K defines a bijection between the sets of isomorphism classes of simple uniformly bounded Harish-Chandra g-modules and simple uniformly bounded Harish-Chandra g0-modules. Theorem 5.6 reduces the classification of simple bounded g-modules to the even part of g. It follows immediatelyb from the following lemma. b Lemma 5.7. Let M be a simple bounded weight g-module ofb level zero. Then the set of b g1 g1-invariants Mb = m M g1m = 0 is a nonzero simple bounded weight g0-module g1 { ∈ | } and M ∼= V (Mb ). b b b g1 b Proof. It is enough to show that the vector space Mb is nonzero as the rest of the proof will follow by standard arguments. Assume that all dimensions are bounded by d 1, and fix some odd root α of g . ≥ 1 Without loss of generality assume that we have a weight vector v = 0 such that xα+sδv =0 for all s =0,... 2d 1. Suppose w = x v = 0, and consider6 the vectors − α+2dδ 6 v, xδx−δv,...,xdδx−dδv.

Since these vectors are linearly dependent, there exists l d such that xlδx−lδv = l−1 ≤ i=1 cixiδx−iδv + c0v. Apply xα+(2d−l)δ on this equation. On the right hand side we obtain 0, and on the left hand side: P xα+(2d−l)δ xlδx−lδv = xα+2dδx−lδv = x−lδxα+2dδv. Thus x x v = x w = 0. From this we conclude that x w = 0 for all s 2d. −lδ α+2dδ −lδ α+sδ ≤ Now, choose the smallest N Z>2d such that xα+Nδw = 0, and consider the linearly dependent elements of M ∈ 6

xldδxα+Nδw, xl(d−1)δxα+(N+l)δ w,...,xlδxα+(N+l(d−1))δ w, xα+(N+ld)δw,

where 1 l d was ﬁxed above. Note that xα+(N+lm)δw = 0 for all 0 m d, since ≤ m≤ 6 ≤ ≤ otherwise x−lδxα+(N+lm)δw equals xα+Nδw up to a nonzero scalar and hence xα+Nδw = 0, which is a contradiction. Assume that for 0 k d holds ≤ ≤ k−1

xl(d−k)δxα+(N+lk)δw = cixl(d−i)δ xα+(N+il)δw i=0 X k and apply x−lδ on this equation. Since c acts trivially on M, x−lδw = 0 and

xl(d−i)δxα+(N+(i−k)l)δ w =0 k for all i

′ ′ xl(d−k)v = 0 for some 0 k

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Departamento de Matematica,´ Instituto de Cienciasˆ Exatas, UFMG, Belo Horizonnte, Minas Gerais, Brazil Email address: [email protected] Instituto de Matematica´ e Estat´ıstica, Universidade de Sao˜ Paulo, Sao Paulo, Brazil, and International Center for Mathematics, SUSTech, Shenzhen, China Email address: [email protected] Instituto de Matematica´ e Estat´ıstica, Universidade de Sao˜ Paulo, Sao Paulo, Brazil Email address: [email protected]