WEYL MODULES AND LEVI SUBALGEBRAS GHISLAINFOURIER 3 1 0 Abstract. ForasimplecomplexLiealgebra offiniterankandclassical type,wefixatrian- 2 gular decomposition and considerthesimpleLevisubalgebras associated toclosed subsetsof roots. We study the restriction of global and local Weyl modules of current algebras to this n Levisubalgebra. WeidentifynecessaryandsufficientconditionsonapairofaLevisubalgebra a J and a dominant integral weight, such that the restricted module is a global (resp. a local) Weyl module. 3 ] T R Introduction . h t a Let g be a finite-dimensional, simple complex Lie algebra of classical type and we fix a trian- m gular decomposition, denoting h the fixed Cartan subalgebra and R the set of roots. We fix [ a subset R ⊆ R, closed under addition and scalar multiplication and consider the associated a subalgebra a, that is the subalgebra generated by the root spaces g , α ∈ R . Then a is the 2 α a v Levi subalgebra of a reductive Lie subalgebra r ⊆ g, having the same Cartan subalgebra as 5 g. We denote π the induced map h∗ ։ h∗, we obtain also an induced map for the dominant a 0 integral weights. 5 6 . 2 LetV(λ) denotethesimplefinite-dimensionalg-moduleof highestweight λ, then weobtain by 1 restricting the module structure a module for a. The a-module generated through the highest 2 weight vector is isomorphic to the a-module V(π(λ)). 1 : By proceeding in thesame way in modularrepresentation theory, starting with the Weyl mod- v ule of highest weight λ of g one obtains also the Weyl module of highest weight π(λ) for a. i X In this paper we will study the behaviour of local and global Weyl modules for the current r algebra under this construction. a Letg⊗C[t]thecurrentalgebraassociatedtog,thisisaLiealgebraequippedwithaLiebracket induced by [x⊗p,y ⊗q] := [x,y]⊗pq. Simple finite-dimensional modules are parametrized by finitely supported functions ψ on C with values in the dominant weights. It was proven in [ER93], that any simple finite-dimensional V(ψ) is isomorphic to the tensor product of evaluations modules (at maximal ideals of C[t]) . Similar to the classical case, the a⊗C[t] module generated through the highest weight vector of V(ψ) is simple and isomorphic to V(π(ψ)) (Lemma 2). Thecategory of finite-dimensionalmodulesis notsemi-simple, soin[CP01]local Weyl modules were defined in analogy to modular representation theory. To any simple module V(ψ) they associated a finite-dimensional module Wg(ψ), that satisfies a certain ”maximality” property. Date: January 4, 2013. G.F. is partially theDFGpriority program 1388 ”Representation Theory”. 1 2 FOURIER The study of these local Weyl modules initiated a serie of papers, computing their dimensions and g-characters ([CP01], [CL06], [FL07], [Nao12]). Local Weyl modules are in particular interesting, since their characters are proven to be characters of certain simple modules of the quantum affine algebra. In [CP01], to each dominant integral weight λ, a so-called global Weyl module Wg(λ) was de- fined, a projective object in the category of integrable g⊗C[t]-modules with weights bounded byλ. Thesemodulesareinfinite-dimensionalandeventhemultiplicity ofthesimpleg-modules in a decomposition is infinite. It was shown that they are also right modules for a certain sym- metric algebra Ag, a polynomial ring in finitely many (determind by λ) variables. Any local λ Weyl module can be obtained by factorizing a global Weyl module at a maximal ideal of Ag. λ We shall remark, that these maximal ideals are also parametrized by finitely supported func- tions on C. In the aforementioned serie of papers it was shown that the dimension of the local Weyl module is indepedent of the maximal ideal, this implies that the global Weyl module is a free right Ag-module of finite rank (for more details see the appendix of [FMSa]). λ There are various generalizations of Weyl modules. For instance one can replace C[t] by any commutative associative unital algebra over the complex numbers ([FL04], [CFK10]), or consider hyper loop algebras ([JdM07]), consider twisted current algebras or more general equivariant map algebras ([CFS08], [FKKS12], [FMSb], [FMSa], [FK]). The importance of the global Weyl module might be seen in various applications, for example the character gives a q-Whittaker function (a solution to the q-Toda integrable system [BF]) or its applications to symmtric functions ([CL12]) or the BGG-property ([BCM12]). We will consider in this paper the restriction of local and global Weyl modules to the simple Levi subalgebra a associated to a closed subset of roots of g. For a pair (a,λ) we introduce the properties local admissible and global admissible (Definition 1.4), and we will see that any global admissible pair is local admissible (Proposition 1.4). We will prove the main theorem of the paper Theorem 1. Let g be a simple, finite-dimensional complex Lie algebra of classical type, a a simple Levi subalgebra and let λ be a dominant integrable weight for g. (1) Let ψ be a finitely supported function of weight λ, then Wa(π(ψ)) ֒→ Wg(ψ) if and only if (a,λ) is local admissible. (2) Wa(π(λ)) ֒→ Wg(λ) if and only if (a,λ) is global admissible. We will prove part (i) of the theorem by reduction to thecase whereλ is a fundamentalweight of g and then calculate explicitly the a-decomposition of the restricted module, showing that it coincides with the decomposition of the local Weyl module. Part (ii) is proven by showing that Aa is a natural subalgebra in Ag and then showing that the restricted module is a free π(λ) λ Aa of the same rank as the global Weyl module Wa(π(λ)). π(λ) It would be interesting to study the restriction to Levi subalgebras also in the case of gen- eralized current algebras, that is replacing C[t] by some algebra A. Not all proofs presented in this paper can be generalized, since in general the dimension of the local Weyl modules is not known and the global Weyl module is not free. The statement for fundamental local Weyl modules (Lemma 4) is remains true in this more general setting (if A is finitely generated). WEYL MODULES AND LEVI SUBALGEBRAS 3 This implies the stament for local Weyl modules (Theorem 8) is true for regular points in A λ (for details, we refer to [CFK10]). Since local and global Weyl modules are characterized by homological properties ([CFK10]) it would be interesting to obtain the results of this paper by using these properties instead of using dimension arguments. This might lead a way to obtain the results for generalized current algebras. In modular representation theory, the results on Weyl modules can be obtained by using argu- mentsfromalgebraicgeometry. Itmightbeusefultoadaptsomeofthemethodstogeneralized current algebras. Another interesting question is whether a g⊗C[t]-module is a local/global Weyl module for g ⊗ C[t] if it is a local/global Weyl module for a ⊗ C[t] for every non-trivial simple Levi subalgebra a, this will be discussed elsewhere. The paper is organized as follows: In Section 1 we fix basic notations, give a complete list of the simple subalgebras of g that will be considered here and introduce the properties local and global admissible. In Section 2 we recall some results on finite-dimensional simple module of g⊗C[t] and analyze their behaviour under restriction. In Section 3 we recall some results on local and global Weyl modules. Part (i) of the main theorem is proven in Section 4, while part (ii) is proven in Section 5. 1. Preliminaries 1.1. Let g be a simple, finite-dimensional complex Lie algebra of classical type, we fix a triangular decomposition of g n+⊗h⊗n−. We denote the set of (positive) roots of g by R (resp. R+). The simple roots of g are denoted α , i ∈ {1,...,n} =:I, the set of simple roots Π. i The (dominant) integral weights are denoted P (resp. P+). The fundamental weights of g are denoted ω , i ∈ I. We have i g = h⊕ g . α α∈R M For a subset R ⊆ R, closed under addition and scalar multiplication, we denote a the Lie a algebra generated by the root spaces corresponding to R+: a a := [g ,g ] g . α −α α αX∈Ra αM∈Ra Then a is the Levi subalgebra of the reductive subalgebra a + h and a is semi-simple by construction. We have an induced triangular decomposition n+⊕h ⊕n− a a a where n± ⊆ n± and h ⊆ h. The simple roots β , j ∈ {1,...,s} =: J, and the set of simple a a j rootsΠ (formoredetailsseeSection1.2). Recall, thattheLevisubalgebrasweareconsidering a are more general than Levi subalgebras associated to parabolic subalgebras. The (dominant) integral weights are denoted P ,P+, while the fundamental weights of a a a denoted τ , j ∈ J. j 4 FOURIER Since h ⊆ h and R+ ⊂ R+ we have an induced maps π : h∗ ։ h∗, π : P −→ P and a a a a π :P+ −→ P+. a 1.2. In the remaining of the paper we will assume that a is a simple Levi algebra and hence we shall give a complete list of all simple Levi subalgebras obtained from closed subsets of R. First we will give a list of the positive roots of g (see [Car05]): • If g is of type A , then n R+ = {ε −ε |1 ≤ i < j ≤n+1}. i j • If g is of type B , then the long roots are n (R+)ℓ = {ε ±ε |1 ≤ i< j ≤ n} i j and the short positive roots are (R+)s = {ε |1 ≤ i ≤ n}. i • If g is of type C , then the long roots are n (R+)ℓ = {2ε | 1 ≤ i≤ n} i and the short positive roots are (R+)s = {ε ±ε | i1 ≤ i< j ≤ n}. i j • If g is of type D , then the roots are n R+ = {ε ±ε | 1≤ i < j ≤ n}. i j The following propositions give a description of all possible sets of simple roots Π of simple a Levi subalgebras. The proofs are omitted, since the results are obtained by a simple case by case consideration. Proposition. Let g be of type A , a ⊆ g a simple Lie subalgebra of rank s, then n Π = {ε −ε ,ε −ε ,...,ε −ε } a i1 i2 i2 i3 is is+1 for some 1 ≤ i < i < ... < i < i ≤ n+1. Especially a is of type A . 1 2 s s+1 s Proposition. Let g be of type B and a⊆ g a simple Lie subalgebra of rank s, then there are n several cases: (1) If ε ∈ Π for some i ∈ I, then a is of type B and ε is the unique simple short root. i a s i Then Π = {ε −ε ,...,ε −ε ,ε } a is−1 is−2 i1 i i (2) If {ε + ε ,ε − ε } ⊂ Π for some i < j, then a is of type D and ε + ε ,ε − ε i j i j a s i j i j correspond to the spin nodes. Then Π = {{ε −ε ,...,ε −ε ,ε +ε ,ε −ε } a is−2 is−3 i1 i i j i j (3) In all other cases a is of type A : s (a) If ε +ε ∈Π for some i < j, then i j a Π = {ε −ε ,...,ε −ε ,ε +ε ,ε −ε ,...,ε −ε } (1.1) a iℓ iℓ−1 i1 i i j j1 j js−ℓ−1 js−ℓ−2 for some 1 ≤ iℓ < ... < i1 < i,1 ≤ js−ℓ−1 < ... <j1 < j and ik 6= jk′ for all k,k′. WEYL MODULES AND LEVI SUBALGEBRAS 5 (b) In the remaining case Π = {ε −ε ,...,ε −ε } (1.2) a i1 i2 is is+1 where 1 ≤ i < ... < i < i ≤ n. 1 s s+1 Proposition. Let g be of type C and a ⊆ g a simple Lie subalgebra of rank s, then there are n several cases: (1) If 2ε ∈ Π for some i∈ I, then a is of type C and 2ε is the unique simple long root. i a s i (2) In all other cases, a is of type A : s (a) If ε +ε ∈Π for some i < j, then Π is of the same form as (1.1). i j a a (b) In the remaining case, Π is of the same form as (1.2). a Proposition. Let g be of type D and a ⊆ g a simple Lie subalgebra of rank s, then there are n several cases (1) If {ε +ε ,ε −ε } ⊂ Π for some i < j ≤ n, then a is of type D and ε +ε ,ε −ε i j i j a s i j i j correspond to the spin nodes. Then Π = {{ε −ε ,...,ε −ε ,ε +ε ,ε −ε } a is−2 is−3 i1 i i j i j (2) In all other cases, a is of type A : s (a) If β +β ∈ Π for some i < j, then Π is of the same form as (1.1). i j a a (b) In the remaining case Π is of the same form as (1.2). a 1.3. We recall some notations from representation theory. Let V be a finite-dimensional g-module, then V decomposes into its weight spaces with respect to the h-action V = V τ τ∈P M where V = {v ∈ V | h.v = τ(h).v for all h ∈ h}. τ The category of finite-dimensional g-module is semi-simple, every object decomposes into the direct sum of simple modules. The simple finite-dimensional modules are parametrized by dominant integral weights V(λ) ↔ λ ∈ P+. Even more dimV(λ) = 1, we denote a generator of this space v . We have further λ λ U(g)/I ∼=V(λ) : 17→ v λ λ where I is the left ideal generated by λ n+ ; h−λ(h) ; (x−)λ(hα)+1 : h∈ h,α ∈ R+. α Leta⊆ gaLiesubalgebraasdefinedinSection1.1,thenthesimplefinite-dimensionalmodules are parametrized by P+ and we denote, by abuse of notation, for µ ∈ P+ the corresponding a a simple a-module V(µ). Then we have Proposition. Let λ ∈ P+, then U(a).v ∼= V(π(λ)) λ a Thisfollows immediately sinceV(λ) isone-dimensionalandthecategory offinite-dimensional λ a-modules is semi-simple. 6 FOURIER 1.4. Here we will introduce the properties of tuples (a,λ) whose implications will be stud- ied in the remaining of the paper. Definition. Let λ = m ω ∈ P+, we call the pair (a,λ) global admissible if i∈I i i m 6=P0 ⇒ ∃j ∈ J : π(ω ) = τ , a fundamental weight for a. i i j The pair (a,λ) is called local non-admissible if (1) either a is of type B , s > 1, ǫ +ǫ is the unique simple short root of B and ∃ i ≤ s i j s k ≤ n−1 such that m 6= 0, k (2) or g is of type C , a is of type A , and ǫ +ǫ ∈ Π and there exists k ∈ I with m 6= 0 n s i j a k and π(ω ) is not a fundamental weight of a. k In all other cases (a,λ) is called local admissible. We can immediately relate this two properties: Proposition. If (a,λ) is global admissible, then (a,λ) is local admissible. Proof. We have to check the case where a is of type B , ǫ +ǫ is the unique simple short root. s i j Then for all i ≤ k ≤ n−1 : π(ω ) = 2τ and hence not a fundamental weight. This gives the k s proposition. (cid:3) Obviously local admissible does not imply global admissible. 1.5. We want to study modules for the current algebra associated to g, that is the Lie algebra g⊗C[t], where the Lie bracket is induced by [x⊗p,y⊗q]= [x,y] ⊗pq g for x,y ∈ g,p,q ∈ C[t]. These current algebra do appear for example as subalgebras of affine Kac-Moody algebras ([Kac90]). We have an inclusion of Lie algebras g ֒→ g⊗C[t] : x 7→ x⊗1. The maximal spectrum of C[t] is C, hence every maximal ideal m corresponds to a unique c∈ C. We have an induced surjective map of Lie algebras g⊗C[t]։ g⊗C[t]/m∼= g : x⊗p 7→ x⊗p(c). Let V be a g-module and m a maximal ideal of C[t] (c the corresponding complex number), then we denote evmV = evcV the induced module obtained by the evaluation at the maximal ideal m. WEYL MODULES AND LEVI SUBALGEBRAS 7 2. Simple finite-dimensional modules We will recall some results on simple finite-dimensional modules for g⊗C[t]. Denote E the monoid of finitely supported functions on C with values in P+ E = {ψ : C−→ P+||supp(ψ)| < ∞} where supp(ψ) = {a ∈ C | ψ(a) 6= 0}. We set wt(ψ) = ψ(a) and for given λ ∈ P+: a∈supp(ψ) P Eλ = {ψ ∈ E| wt(ψ) = λ}. To each ψ ∈ E we associated a finite-dimensional g⊗C[t]-module V(ψ) = ev V(ψ(a)). a a∈suppψ O Recall that the order of the tensor product does not matter, hence every ψ determines an isomorphism class of modules. Let v be a generator of V(ψ(a)) (a one-dimensional space, a ψ(a) so the generator is unique up to scalars). We denote v = ⊗ v , ψ a∈supp(ψ) a this is a generator of V(ψ) . We will use this notation throughout the paper. wt(ψ) Theorem 2. ([ER93],[Lau10], [NSS12], [CFK10]) Let ψ ∈ E, then V(ψ) is simple g⊗C[t]- module. Further if V is a simple finite-dimensional g⊗C[t]-module, then there exists a unique ψ ∈ E such that V ∼= V(ψ). So E parametrizes the finite-dimensional simple g⊗C[t]-modules up to isomorphism. Similar we can define E , Eµ for µ ∈ P+. The map π : P+ −→ P+ induces a map a a a a π : E −→ E : ψ 7→ π◦ψ. a For all λ ∈ P+, this map restricts to π : Eλ −→ Eπ(λ). a Lemma. Let ψ ∈ Eλ, then U(a⊗A).v ∼= V(π(ψ)) ψ where v ∈ V(ψ) . ψ λ Proof. Let supp(ψ) = {a ,...,a }, then by definition of V(ψ): 1 k k g⊗ (t−a )C[t] .V(ψ) = 0. i ! i=1 Y This implies that V(ψ) is a simple module for the semi-simple Lie algebra k k g⊗C[t]/ (t−a ) ∼= g⊗(C[t]/(t−a )). i i ! i=1 i=1 Y M 8 FOURIER We have k V(ψ) = ev V(ψ(a )) ai i i=1 O and if v is a generator of V(ψ(a )) , then i i ψ(ai) k k k U(g⊗C[t]/ (t−a )) .⊗v = U(g⊗C[t]/(t−a )).v ∼= ev V(ψ(a )) i i i i ai i ! i=1 i=1 i=1 Y O O This implies that (U(a⊗C[t])).⊗v = U(a⊗C[t]/ (t−a )C[t]) .⊗v = U(a⊗C[t]/(t−a )).v . i i i i i The lemma follows since(cid:16) Y (cid:17) O k U(a⊗C[t]/(t−a )).v ∼= ev V(π(ψ(a ))) ∼= V(π(ψ)). i i ai i i=1 O O (cid:3) We shallremarkhere,thatfor given λ ∈ P+, V(λ)is afinite-dimensionala-module, sothere is a decomposition V(λ) ∼=a V(τ)⊕cτλ. τM∈Pa+ This gives a decomposition formula of V(ψ) into simple a⊗C[t]-modules. Unfortunately, a decomposition formula of V(λ) as a a-module is not known in general. The following proposition is essential in the proof of Theorem 1(ii). Proposition. Let (a,λ) be global admissible and φ ∈ Eπ(λ). Then there exists ψ ∈ Eλ such a that V(φ) ∼= U(a⊗C[t]).v , π(ψ) = φ a ψ and the map π : Eλ −→ Eπ(λ) is surjective. a Proof. Let λ = m ω and π(λ) = n τ . Suppose we have a decomposition of i∈I i i j∈J j j λ = µ +...+µ , with µ ∈ P+. Then (a,λ) is global admissible if and only if (a,µ ) is global 1 kP ℓ P ℓ admissible for all ℓ ∈{1,...,k}. Since (a,λ) is global admissible we can find for all j ∈ J, such that n 6= 0, a i ∈ I with j j π(ωij) = τj. Then ij 6= ij′ if j 6= j′. This implies that if π(λ) = ν +...+ν is a decomposition into dominant weights for a, then 1 k there exists µ ,...,µ such that 1 k λ = µ +...+µ and π(µ )= ν for all 1 ≤ℓ ≤ k. (2.1) 1 k ℓ ℓ Let φ ∈ Eπ(λ). We have to show that there exists ψ ∈ Eλ such that π(ψ) = φ. We can write a φ= φ , where a∈supp(φ) a P φ(a) if b = a φ (b) := a (0 else WEYL MODULES AND LEVI SUBALGEBRAS 9 It is clearly enough to find preimages for all φ . By (2.1), for all a ∈ supp(φ) there exists a µa ∈ P+ such that λ = µa and π(µa)= φ(a). a∈supp(φ) X We define now ψ ∈ Eµa by a µa if b = a ψ (b) := a (0 else Then π(ψ ) = φ and wt(ψ ) = λ. a a a a∈supp(φ) X We set ψ := ψ . a a∈supp(φ) X Then we have ψ ∈ Eλ and π(ψ) = φ. But then with Lemma 2 V(φ) ∼=a⊗C[t] U(a⊗C[t]).vψ and the proposition follows. (cid:3) Clearly, if (a,λ) is not global admissible, then the map π :Eλ −→ Eπ(λ) is not surjetive. a 3. Global and local Weyl modules In this section we recall the definitons and some facts on global and local Weyl modules. 3.1. Following [CP01] we define Definition. Letλ ∈ P+,thentheg⊗C[t]-moduleWg(λ)generated throughanon-zerovector w subject to the relations λ n+⊗C[t].w = 0 , (h⊗1).w = 0 , (x−⊗1)λ(hα)+1.w = 0 λ λ α λ for all h ∈h and α ∈ R+, is called the global Weyl module of weight λ. We shall make a couple of remarks here Remark. We shall make a couple of remarks here (1) Inthedefinition,C[t]canbereplacedbyacommutative, associativeunitalalgebraover C. One may see for example [CFK10], [FL04] to get an insight about the difficulties and differences to the classical situation that one faces here. (2) Gobal Weyl modules can be also defined for twisted loop algebras ([FMSb]) or more general for equivariant map algebras. ([FMSa]) (3) By construction, Wg(λ) is an integrable g-module but infinite dimensional, even the multiplicity of every simple g-module appearing in a decomposition is infinite. (4) Wg(λ)isprojectiveinthecategoryofintegrableg⊗C[t]-moduleswithweightsbounded above by λ (see [CFK10]). 10 FOURIER (5) Wg(λ) plays an important role in the context of q-Toda integrable systems, namely their characters give so-called q-Whittaker functions (see [BF]). We denote Ag := U(h⊗C[t])/Ann (U(h⊗C[t])). λ wλ Then Ag is a commutative, associative, unital algebra and we can say even more ( [CP01], λ resp [CFK10] for the second part): Theorem 3. If λ = m ω , then i∈I i i P Ag ∼= Smi(C[t]) λ i∈I O where Smi(C[t]) denotes the mi-th symmetric algebra of C[t]. Further Ag ∼=U(h⊗C[t])/I λ where I = AnnU(h⊗C[t])(vψ). ψ∈Eλ \ For more on this algebra, the interested reader may also see [CFK10], [CL12]. Wg(λ) admits the structure of a (g,Ag)-bimodule, where the right action is given by λ uw .h⊗a := u(h⊗a)wλ λ for u∈ U(g⊗C[t]),h⊗a ∈a⊗C[t]. With this we have the following theorem which follows from results due to [CP01] for g ∼= sl , 2 [CL06] for g ∼= sl , [FL06] for g of simply-laced type and [Nao12] for classical g. n+1 Theorem 4. Let λ ∈P+, then Wg(λ) is a free right Ag-module of finite rank. λ 3.2. The global Weyl module Wg(λ) induces a functor from the category of Ag-modules λ to the category of integrable g⊗C[t]-modules with weights bounded above by λ: M 7→ Wg(λ)⊗Ag M ; f 7→ 1⊗f λ Since Ag is a polynomial ring in finitely many variables, so especially finite generated, any λ one-dimensional Ag-module is isomorphic to Ag/m for a uniquely determind maximal ideal λ λ of Ag. Maximal ideals of Ag are parametrized by Eλ (see [CFK10] for details), so to any λ λ maximal ideal m of Ag there exists a finitely supported function ψ of weight λ. λ Definition. Let λ ∈ P+ and ψ ∈ Eλ, denote m be the corresponding maximal ideal of Ag. λ The g⊗C[t]-module Wg(λ)⊗Ag Agλ/m λ is called the local Weyl module associated to ψ and denoted Wg(ψ). Remark. We shall make a couple of remarks here: