A generalisation of the category O of Bernstein-Bernstein-Gelfand
aa r X i v : . [ m a t h . R T ] D ec A generalization of the category O of Bernstein–Gelfand–Gelfand Guillaume Tomasini
IRMA, CNRS et Université de Strasbourg,7 rue René Descartes, 67084 STRASBOURG CEDEX
Résumé
L’étude des représentations irréductibles d’une algèbre de Lie simple définie sur le corps des nombres complexesa conduit Bernstein, Gelfand et Gelfand a introduire une catégorie qui fournit un cadre naturel pour les modulesde plus haut poids. Le but de cette note est de présenter une construction d’une famille de catégories généralisantcelle de Bernstein–Gelfand–Gelfand. Nous décrivons les modules simples de certaines de ces catégories. Cetteclassification permet de montrer que ces catégories sont semi–simples.
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Abstract
In the study of simple modules over a simple complex Lie algebra, Bernstein, Gelfand and Gelfand introduced acategory of modules which provides a natural setting for highest weight modules. In this note, we define a familyof categories which generalizes the BGG category. We classify the simple modules for some of these categories. Asa consequence we show that these categories are semisimple.
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1. Weight modules and Generalized Verma Modules
Let g denote a simple Lie algebra over C and U ( g ) denote its universal enveloping algebra. Let h be afixed Cartan subalgebra and denote by R the corresponding set of roots. For α ∈ R we denote by H α ∈ h the corresponding co–root and by g α the rootspace for the root α . We will denote by ∆ a set of simpleroots. We write R + for the corresponding set of positive roots. For a subset θ ⊂ ∆ , we denote by h θ i theset of all roots which are linear combination of elements of θ and set h θ i ± = h θ i ∩ R ± . We consider thefollowing subalgebras of g : l θ = h ⊕ X α ∈h θ i g α , n ± θ = X α ∈R ± −h θ i ± g α . Email address: [email protected] (Guillaume Tomasini).
Preprint submitted to the Académie des sciences November 9, 2018 he subalgebra p θ = l θ ⊕ n + θ is called the standard parabolic subalgebra associated to θ and l θ is thestandard Levi subalgebra. The later is a reductive algebra. Its semisimple part is denoted l ′ θ . If θ = ∅ ,then l ∅ = h and we simply write n + instead of n + ∅ .We denote by M od ( g ) the category of all g –modules. We will investigate some full subcategories of M od ( g ) for which we will describe the simple modules. A module M is a weight module if it is finitelygenerated, and h –diagonalizable in the sense that M = ⊕ λ ∈ h ∗ M λ , M λ = { m ∈ M : H · m = λ ( H ) } , with weight spaces M λ of finite dimension. We will denote by M ( g , h ) the full subcategory of M od ( g ) consisting of all weight modules. This category has been studied by several authors (e.g. [7], [8]). Thiscategory also appears as a particular case of generalised Harisch–Chandra modules (see [14] for a defini-tion).The Bernstein–Gelfand–Gelfand category O is the full subcategory of M ( g , h ) whose objects are n + –finite (a module M is n + –finite if for all m ∈ M the set U ( n + ) m is a finite dimensional vector space). Thecateory O was introduced by Bernstein, Gelfand and Gelfand (see [2]). There they desbribe the simplemodules and the structure of the category itself, namely they study the projective modules in O and givea certain correspondence for multiplicities (the so called BGG–correspondence). For a review of theseresults we refer the reader to [10]. In order to discuss the simple modules in M ( g , h ) we need to recallsome well known facts about generalised Verma modules.A l θ –module can be made into a p θ –module by letting n + θ act trivially. For such a module N we definethe generalised Verma module (GVM) V ( θ, N ) by V ( θ, N ) = U ( g ) ⊗ U ( p θ ) N. For any g –module M wedefine the l θ –module M n + θ := { m ∈ M | n + θ m = 0 } . Let us recall the following classical facts about GVMs:
Proposition 1.1 (i) If N is a simple l θ –module, the module V ( θ, N ) admits a unique simple quotient,denoted by L ( θ, N ) . The module L ( θ, N ) is called the simple g –module induced from ( l θ , N ) .(ii) If M is a simple g –module such that M n + θ = { } , then M ∼ = L ( θ, M n + θ ) . We refer to [7, proposition . ] for a proof. We refer to [13] for a more detailed discussion of GVM’s.To give the classification of simple weight modules, we need one more ingredient: the so–called cuspidal modules.
2. Cuspidal modules
Let M be a weight module. A root α ∈ R is said to be locally nilpotent (with respect to M ) if X α ∈ g α acts by a locally nilpotent operator on the whole module M . It is said to be cuspidal if X α acts injectivelyon the whole module M . We denote by R N ( M ) the set of locally nilpotent roots and by R I ( M ) the setof cuspidal roots. We shall simply denote R N and R I when the module M is clear from the context.It is known that for a simple weight module R = R N ⊔ R I (see [7, lemma . ]). A weight module iscalled cuspidal if R = R I . Set R Ns = { α ∈ R N : − α ∈ R N } , R Na = R N − R Ns . We define R Is and R Ia the same way. Recall the following theorem: Theorem 2.1 (Fernando [7, theorem . ], Futorny [8]) Let M be a simple weight module. Then M is induced from a cuspidal simple module of some Levi subalgebra of g . More precisely there is a set a simple roots ∆ and a subset θ ⊂ ∆ such that h θ i = R Is and R + − h θ i + = R N . Then M n + θ is a simple cuspidal module for l θ and M ∼ = L ( θ, M n + θ ) .The theorem of Fernando reduces the classification of simple weight g –modules to the classification ofsimple cuspidal weight modules for reductive Lie algebras. By standard arguments this reduces to theclassification of simple cuspidal modules for simple Lie algebras. A first step towards this classification isgiven by the following theorem: 2 heorem 2.2 (Fernando [7, theorem . ]) Let g be a simple Lie algebra. If M is a simple cuspidal g –module, then g is of type A or C The classification of simple cuspidal modules was then completed in two steps. In the first step Brittenand Lemire classified all simple cuspidal modules of degree (see [5]) where deg ( M ) = sup λ ∈ h ∗ { dim ( M λ ) } . Britten and Lemire, and later Benkart, Britten and Lemire, have classified all simple modules of degree when g is of type A or C (see [1]). These modules will play a key role in our theorem 3.2 below.Later Mathieu gave the full classification of simple cuspidal modules of finite degree greater than byintroducing the notion of a coherent family (see [12]).
3. The category O S,θ
Now we define a family of full subcategories of M ( g , h ) . Fix a set of simple roots ∆ . Fix two subsets S and θ of ∆ such that θ ⊂ S . The category O S,θ is the full subcategory of M ( g , h ) whose objects are themodules satisfying the following conditions : ( O As a l θ –module, M is a direct sum of simple highest weight modules. ( O For α ∈ h S − θ i , the element X α ∈ g α is cuspidal for M . ( O The module M is n + S –finite.Notice than in the case S = θ = ∅ , we recover the definition of the category O . Other generalisations ofcategory O can be recovered for particular choices of ( θ, S ) (see [15], [13]). We sometimes write O S,θ ( g ) toemphasize the Lie algebra. Now we list some easy properties of O S,θ which are generalisations of analogousproperties of O . Proposition 3.1
Let
M, N ∈ O
S,θ .(i) Then M ⊕ N is in O S,θ . Every submodule and every quotient of M is again in O S,θ .(ii) The category O S,θ is abelian, and every module in O S,θ is noetherian and artinian.(iii) The category O S,θ is Z ( g ) –finite (where Z ( g ) is the center of U ( g ) ).(iv) The simple modules in O S,θ are modules of the form L ( S, N ) where N is a simple module in O S,θ ( l S ) . Being artinian and noetherian, every module M in O S,θ admits a finite Jordan–Hölder series whosequotients are of the form L ( S, N ) (see [11]). This allows us to define the multipicity of L ( S, N ) in anyJordan–Hölder series of M . From proposition 3.1 ( iv ) , the description of the simple modules in O S,θ ( g ) first requires to consider the case where S = ∆ .In the sequel we assume S = ∆ . We shall only consider a simple Lie algebra g and restrict ourselvesto the case θ = ∅ and θ = ∆ . We label the simple roots as in [3]. First, from theorem 2.1, every simplemodule in O ∆ ,θ is of the form L (∆ − θ, C ) for some simple cuspidal l ∆ − θ –module C . Such a module isdefined by a simple cuspidal l ′ ∆ − θ –module C ′ and a scalar action of the center z of l ∆ − θ . We have to findthose C for which L (∆ − θ, C ) satisfies condition ( O of the definition of the category O ∆ ,θ . Theorem 3.2
Assume ( g , ∆ − θ ) is not one of the following: ( B n , { e } ) , ( D n , { e } ) , ( D n , { e n − } ) , ( D n , { e n } ) , ( E , { e } ) , ( E , { e } ) , ( E , { e } ) . Then we have(i) If L (∆ − θ, C ) is a simple module in O ∆ ,θ ( g ) , then g is of type A n or is of type C n , ∆ − θ is aconnected subset of ∆ and C is a simple cuspidal module of degree .(ii) If g = sl n +1 and ∆ − θ is any connected subset of ∆ other than { e } and { e n } , then the simplemodules in O ∆ ,θ ( g ) are modules of degree . Conversely, any infinite dimensional simple sl n –moduleof degree is a simple object in some category O ∆ ,θ ( sl n ) .(iii) If g = sp n , then there are simple modules in O ∆ ,θ ( g ) if and only if ∆ − θ ⊃ { e n } . In this case, allthe simple modules in O ∆ ,θ ( g ) are modules of degree . Conversely, any infinite dimensional simple sp n –module of degree is a simple object in some category O ∆ ,θ ( sp n ) . g is of type A or C and that θ is neither empty nor equal to ∆ . When g = A n we also assume that ∆ − θ is different from { e } and { e n } . When the category O ∆ ,θ is non empty,we want to describe its structure. First we note the following fact: if F is a finite dimensional g –moduleof dimension greater than and M ∈ O ∆ ,θ is simple, then F ⊗ M does not belong to the category O ∆ ,θ in general. The proof of this fact requires some results of Britten and Lemire [6, section ]. Thereforethe category O ∆ ,θ has a structure very different from the case of the category O . In fact, we prove thefollowing Theorem 3.3
Let M and N be two simple modules in O ∆ ,θ . Then Ext O ∆ ,θ ( M, N ) = 0 . As a corollary we get:
Corollary 3.4
The category O ∆ ,θ is semisimple. Remark that the result holds trivially when θ = ∆ (this is in fact a part of the definition of thecategory). Note that when θ = ∅ and g is of type C , Britten, Khomenko, Lemire, Mazorchuk haveproved the semisimplicity of O ∆ ,θ in [4]. The result does not hold when θ = ∅ and g is of type A . In[9], Grantcharov and Serganova have described the indecomposable modules in this case. The proof oftheorem 3.3 uses the example . of [9].Detailed proofs will be published elsewhere.References [1] G. Benkart, D. Britten, F. Lemire, Modules with bounded weight multiplicities for simple Lie algebras, Math. Z. (2)225 (1997) 333-353.[2] I. N. Bernšte˘ın, I. M. Gel ′ fand, S. I. Gel ′ fand, Differential operators on the base affine space and a study of g -modules,in Lie groups and their representations (Proc. Summer School, Bolyai János Math. Soc., Budapest, 1971) (1975) 21-64.[3] N. Bourbaki, Groupes et Algèbres de Lie, Chap. IV–VI, Hermann, Paris, 1968.[4] D. Britten, O. Khomenko, F. Lemire, V. Mazorchuk, Complete reducibility of torsion free C n -modules of finite degree,J. Algebra 276 (2004) 129-142.[5] D. Britten, F. Lemire, A classification of simple Lie modules having a -dimensional weight space, Trans. Amer. Math.Soc.. (2) 299 (1987) 683-697.[6] D. Britten, F. Lemire, Tensor product realizations of simple torsion free modules, Canad. J. Math., (2) 53 (2001)225–243.[7] S. L. Fernando, Lie algebra modules with finite-dimensional weight spaces. I, Trans. Amer. Math. Soc. (2) 322 (1990)757-781.[8] V. Futorny, The weight representations of semisimple finite dimensional Lie algebras, PhD Thesis, Kiev University,1987.[9] D. Grantcharov, V. Serganova, Cuspidal representations of sl ( n + 1) , arXiv:0710.2682v1.[10] J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category O , vol. 94 of Graduate Studies inMathematics, American Mathematical Society, Providence, RI, 2008.[11] N. Jacobson, Basic Algebra II , W.H. Freeman, 1980.[12] O. Mathieu, Classification of irreducible weight modules, Ann. Inst. Fourier (Grenoble), (2) 50 (2000) 537-592.[13] V. Mazorchuk, Generalized Verma modules, vol. 8 of Mathematical Studies Monograph Series, VNTL Publishers, L ′ viv,2000.[14] I. Penkov, V. Serganova, Generalized Harish-Chandra modules, Mosc. Math. J., (2) (2002) 753-767.[15] A. Rocha-Caridi, Splitting criteria for g -modules induced from a parabolic and the Berňste˘ın-Gel ′ fand-Gel ′ fandresolution of a finite-dimensional, irreducible g -module, Trans. Amer. Math. Soc., (2) 262 (1980) 335–366.-module, Trans. Amer. Math. Soc., (2) 262 (1980) 335–366.