aa r X i v : . [ m a t h . QA ] A ug GROUPS OF EXTENDED AFFINE LIE TYPE
SAEID AZAM, AMIR FARAHMAND PARSA
Abstract.
We construct certain Steinberg groups associated to extended affineLie algebras and their root systems. Then by the integration methods of Kac andPeterson for integrable Lie algebras, we associate a group to every tame extendedaffine Lie algebra. Afterwards, we show that the extended affine Weyl group of theground Lie algebra can be recovered as a quotient group of two subgroups of thegroup associated to the underlying algebra similar to Kac–Moody groups. Introduction
In 1985 K. Saito introduced extended affine root systems (EARS) in order to presenta suitable geometric space for modeling the universal deformation of a simple ellipticsingularity (see [21]). A few years later, two mathematical physicists, Høegh-Krohn andToresani, constructed certain Lie algebras whose root systems resembled some of theextended affine root systems (see [13]). These algebras were first called quasi-simpleLie algebras but, because of their root systems, later they were called extended affineLie algebras (EALA). These algebras and their root systems are thoroughly studied in[1].Although there is a rather rich literature on EALAs, their associated Weyl groupsand EARSs, there is little about their groups (see [3, 4, 5, 6, 7, 16, 20]). Concerning theconstruction of groups associated to EALAs and EARSs we can notably mention twoworks: [16] and [18]. For affine Kac–Moody algebras, and in general, all Kac–Moodyalgebras there are various methods to associated a group structure to these algebras.In [15] Kac gives a recipe for a group construction associated to integrable
Lie al-gebras by means of representation of Lie algebras. Later on, Tits in [23] associates agroup construction by generators and relations to Kac–Moody root systems similar tothe methods used for finite root systems introduced by Chevalley and Demazure (see[10, 11, 12]).Considering EALAs as a certain generalization of affine Kac–Moody algebras, byfollowing [23], we define certain groups associated to EALAs and EARSs by generatorsand relations. These groups are called
Steinberg . For the simply-laced cases this wasdone in [16] and we generalize the results therein for arbitrary EALAs.By definition, EALAs are integrable in the sense of [15], hence we use Kac andPeterson methods to produce “integration” groups for such EALAs. Furthermore, we
Mathematics Subject Classification.
Key words and phrases. Extended affine, Steinberg, Kac–Moody :This research was in part supported by a grant from IPM and carried out in IPM-Isfahan Branch. find an epimorphism from Steinberg groups to these “integration” groups associatedto a fixed EALA. This epimorphism together with the generalized presentation byconjugations of the extended affine Weyl groups (see [7]) enables us to show thatextended affine Weyl groups can be recovered as a quotient group of two subgroups ofthe “integration” groups. This generalizes similar results in [16] from simply laced toarbitrary types. 2.
General setup
Let C be the field of complex numbers. All vector spaces and algebras are assumedto be over C unless otherwise stated. By h T i , we mean the subgroup generated by asubset T of the ground group or vector space. For a vector space V over R equippedwith a symmetric form ( · , · ), we set V to be the radical of the form, also for R ⊆ V , weset R = R ∩ V and R × = R \ R . For α ∈ V with ( α, α ) = 0, we set α ∨ := 2 α/ ( α, α ),and define the reflection w α ∈ GL ( V ) by w α ( β ) = β − ( β, α ∨ ) α . Sometimes we alsouse the notation h α, β i := ( α, β ∨ ) . Let ( L , ( · , · ) , H ) be a tame irreducible extended affine Lie algebra (EALA). Thismeans that L is a Lie algebra, H is a non-trivial subalgebra of L and ( · , · ) is a symmetricbilinear form on L satisfying (A1)-(A5) below.(A1) ( · , · ) is invariant and non-degenerate,(A2) H is a finite-dimensional Cartan subalgebra of L .Axiom (A2) means that L = P α ∈H ⋆ L α with L α = { x ∈ L | [ h, x ] = α ( h ) x for all h ∈H} , and L = H . Let R be the set of roots of L , namely R = { α ∈ H ⋆ | L α = { }} . Itfollows from (A1)-(A2) that the form ( · , · ) restricted to H is non-degenerate and so itcan be transferred to H ⋆ by ( α, β ) := ( t α , t β ) where t α ∈ H is the unique element sat-isfying α ( h ) = ( h, t α ), h ∈ H . Let R = { α ∈ R | ( α, α ) = 0 } and R × = R \ R . Then R = R ⊎ R × is regarded as the decomposition of roots into isotropic and non-isotropic roots, respectively. Let V := Span R R and V := Span R R .(A3) For α ∈ R × and x ∈ L α , ad( x ) acts locally nilpotently on L .(A4) The Z -span of R in H ⋆ is a free abelian group of rank dim V .(A5) (a) R × is indecomposable,(b) R is non-isolated , meaning that R = ( R × − R × ) ∩ V . The core of an EALA L is by definition, the subalgebra L c of L generated by non-isotropic root spaces. It follows that L c is a perfect ideal of L . When L ⊥ c := { x ∈ L | ( x, L c ) = { }} is contained in the core, L is called tame .Let ( L , ( · , · ) , H ) be an EALA. Let W L be the Weyl group of L ; the subgroup of GL ( H ⋆ ) generated by reflections w α , α ∈ R × . By [1, Theorem I.2.16], the root system R of L is an irreducible extended affine root system in the following sense. Definition 2.1.
Let V be a finite-dimensional real vector space equipped with a non-trivial positive semidefinite symmetric bilinear form ( · , · ), and R a subset of V . Triple( V , ( · , · ) , R ), or R if there is no confusion, is called an extended affine root system (EARS) if the following axioms hold:(R1) h R i is a full lattice in V , ROUPS OF EXTENDED AFFINE LIE TYPE 3 (R2) ( β, α ∨ ) ∈ Z , α, β ∈ R × ,(R3) w α ( β ) ∈ R for α ∈ R × , β ∈ R ,(R4) R = V ∩ ( R × − R × ),(R5) α ∈ R × ⇒ α R .We say that R is irreducible or connected if R × cannot be written as the union of twoof its non-empty orthogonal subsets. Remark 2.2.
One checks that the definition of an irreducible EARS given abovecoincides with the definition of an extended affine root system given in [1, DefinitionII.2.1].Let ( V , ( · , · ) , R ) be an irreducible EARS. It follows that the canonical image ¯ R of R in ¯ V := V / V is a finite root system in ¯ V . The type and the rank of R is defined tobe the type and the rank of ¯ R , and the dimension of V is called the nullity of R . R is called reduced if ¯ R is reduced. Taking an appropriate pre-image ˙ R of ¯ R in V , underthe canonical map V −→ ¯ V , one can see that ˙ R is a finite root system in ˙ V := Span R ˙ R isomorphic to ¯ R . Moreover, one obtains a description of R in the form(2.1) R = R ( ˙ R, S, L, E ) := ( S + S ) ∪ ( ˙ R sh + S ) ∪ ( ˙ R lg + L ) ∪ ( ˙ R ex + E ) , where ˙ R sh , ˙ R lg and ˙ R ex are the sets of short, long and extra long roots of ˙ R , respec-tively, and S , L and E are certain subsets of R , called (translated) semilattices , whichinteract in a prescribed way (see [1, Chapter II] for details). If R is reduced, then in(2.1), ˙ R ex and E are interpreted as empty sets, so(2.2) R = R ( ˙ R, S, L ) = ( S + S ) ∪ ( ˙ R sh + S ) ∪ ( ˙ R lg + L ) . for some reduced finite root system ˙ R . For α ∈ R × , we denote by ˙ α the unique elementin ˙ R associated to α via the description (2.1), namely α = ˙ α + δ α for ˙ α ∈ ˙ V and δ α in S , L or E .We have V = ˙ V ⊕ V . Set ˜ V := V ⊕ ( V ) ⋆ . We extend the form on V to a non-degenerate form on ˜ V by the natural dual pairing of V and ( V ) ⋆ , namely ( λ, σ ) := λ ( σ )for λ ∈ ( V ) ⋆ and σ ∈ V . Let W be the subgroup of GL ( ˜ V ) generated by reflections w α , α ∈ R × . If R is the root system of an extended affine Lie algebra, then W ∼ = W L .We refer to W as the (extended affine) Weyl group of R . Extended affine Weyl groupsare not in general Coxeter groups [14, Theorem 1.1]. However, they enjoy a relatedpresentation called “generalized presentation by conjugation” which will be used in thesequel. We recall briefly this presentation here. Assume that R is reduced and that θ s and θ ℓ are the highest short and highest long roots of ˙ R . One knows that S containsa Z -basis { σ , . . . , σ n } of the lattice Λ := h R i = h S i (see [1, Proposition II.1.11]), andthat if ˙ R lg = ∅ then | Λ / h L i| = k t , where k ∈ { , } (see [1, Chapter II.4]). The integer t is called the twist number of R .For α ∈ R × and σ = P ni =1 m i σ i ∈ R with α + σ ∈ R , we set(2.3) c ( α,σ ) := ( w α + σ w α )( w α w α + σ ) m ( w α w α + σ ) m . . . ( w α w α + σ n ) m n . SAEID AZAM, AMIR FARAHMAND PARSA
Consider a pair( α p , η p = n X i =1 m ip σ i ) ∈ ( { θ s } × n X i =1 Z σ i ) [ ( { θ ℓ } × n X i = t +1 Z σ i ) , and let ǫ p ∈ {± } . The triple { ( ǫ p , α p , η p ) } mp =1 is called a reduced collection if m X p =1 k ( α p ) ǫ p m ip m jp = 0 for all 1 ≤ i < j ≤ n, where k ( θ ℓ ) = 1 for all types, and k ( θ s ) = 3 for type G and k ( θ s ) = 2 for the remainingtypes. Here we have used the convention that if the type of R is simply laced, then θ := θ s = θ ℓ and k ( θ ) = 1. We note from [7, §
2] that if { ( ǫ p , α p , η p ) } mp =1 is a reducedcollection then c ( α p ,η p ) is an element of W . We record here the following fact whichwill be used in the sequel. Theorem 2.3. [7, Theorem 3.7]
The Weyl group W is isomorphic to the group ˆ W defined by generators ˆ r α , α ∈ R × and relations: (i) ˆ r = 1; α ∈ R × , (ii) ˆ r α ˆ r β ˆ r α = ˆ r r α ( β ) ; α, β ∈ R × , (iii) Q mp =1 ˆ c ( α p ,η p ) = 1 for any reduced collection { ( ǫ p , α p , η p ) } mp =1 ,where ˆ c ( α p ,η p ) is the element in ˆ W corresponding to c ( α p ,η p ) , under the assignment w α ˆ w α . The presentation given in Theorem 2.3 is called the generalized presentation byconjugation for W . If W is isomorphic to the presented group determined by (i)-(ii)of Theorem 2.3, then W is said to have the presentation by conjugation . Given anEARS R , there is a computational procedure to decide whether its Weyl group haspresentation by conjugation or not, see [8]. In particular, it is known that all EARS ofnullity ≤ Nilpotent pairs
For the rest of this work, we assume that ( L , ( · , · ) , H ) is a tame extended affineLie algebra and ( V , ( · , · ) , R ) is an irreducible reduced extended affine root system.Whenever, we work with L , we assume R is its root system. We proceed with the samenotation as in Section 2. Definition 3.1.
We say a subset T of R is a subsystem of R if T is an EARS in V T := span R T with respect to the form induced from V on V T . In particular, we say T is a finite or an affine subsystem of R , if T is a finite or an affine root system on itsown. Lemma 3.2.
Let T be an irreducible finite subsystem of R . Then the subalgebra L T of L generated by L ± α , α ∈ T \ { } , is a finite-dimensional simple Lie algebra of type T .Moreover, L T = X α ∈ T \{ } L α ⊕ X α ∈ T [ L α , L − α ] . ROUPS OF EXTENDED AFFINE LIE TYPE 5
Proof.
It follows from [1, Theorem 1.29] that M T := X α ∈ S \{ } L α ⊕ X α ∈ T [ L α , L − α ]is a finite-dimensional simple Lie algebra of type T . Since the roots appearing in L T belong to the root strings of roots in T , and T as a finite root system is closed underroot strings, we conclude that L T ⊆ M T . Clearly M T ⊆ L T . (cid:3) Let T be a subsystem of R and λ ∈ Λ. We set(3.1) T λ := ( T × + Z λ ) ∩ R and R T,λ := T λ ∪ (cid:0) ( T λ − T λ ) ∩ R (cid:1) . Proposition 3.3.
Let T be a finite irreducible subsystem of R and = λ ∈ Λ . Then R T,λ is an affine irreducible subsystem of R . Proof.
Let V λ := span R R T,λ and V λ := V λ ∩ V . Since 2Λ ⊆ S ⊆ R we have, for γ ∈ R × , γ + 2Λ ⊆ ˙ R sh + S + 2Λ ⊆ ˙ R sh + S ⊆ R × if γ is short,and γ + 2Λ ⊆ ˙ R lg + L + 2Λ ⊆ ˙ R lg + L ⊆ R × if γ is long.Thus T × + 2 Z λ ⊆ T λ and so 2 Z λ ⊆ ( T λ − T λ ) ∩ R ⊆ V λ . It follows from this andthe fact that T ∩ V = { } that V λ = R λ , namely the radical of the form restrictedto R T,λ is one-dimensional. Using this and [2, Theorem 2.31], it suffices to show that R T,λ is an irreducible EARS. The irreducibility of R T,λ follows immediately from theirreducibility of T , so it is enough to show that (R1)-(R5) hold. Axioms (R1) and (R2)clearly hold for R T,λ . Axiom (R4) holds as R T,λ = ( T λ − T λ ) ∩ R . Since by assumption R is reduced, (A5) holds. Next, if γ + nλ, γ ′ + n ′ λ ∈ R × T,λ , where γ, γ ′ ∈ T × , then w γ + nλ ( γ ′ + n ′ λ ) = w γ ( γ ′ ) + n ′ λ + ( γ ′ , γ ∨ ) nλ ∈ R ∩ ( T × + Z λ ) = R × T,λ . Thus (R3) holds and the proof is complete. (cid:3)
Let T be an irreducible finite subsystem of R and 0 = δ ∈ R . By Proposition 3.3, R T,δ is an irreducible affine subsystem of R . By a standard argument (see [1, ChapterI]), one can pick an element γ ∈ H ⋆ such that ( γ, α ) = ( γ, γ ) = 0 for α ∈ T and( γ, δ ) = 0. Let d = t γ , the unique element in H which represents γ via the form. Set(3.2) L T,δ : X = α ∈ R T,δ L α + X α ∈ R T,δ [ L α , L − α ] + C d. Lemma 3.4.
Let T be an irreducible finite subsystem of R and δ ∈ Λ . Then L T,δ is anextended affine Lie algebra of nullity with root system R T,δ . Moreover the subalgebra L aT,δ = hL α | α ∈ R × T,δ i ⊕ C d of L T,δ constitutes an affine Lie algebra with root systems R T,δ . Proof.
By [1], we have L T,δ = H T,δ ⊕ X = α ∈ R T,δ L α where H T,δ = P α ∈ T C t α ⊕ C t δ ⊕ C d. From the way d is chosen, the form is non-degenerate (and clearly invariant) on both L T,δ and H T,δ . In particular, we have H = SAEID AZAM, AMIR FARAHMAND PARSA H T,δ ⊕ H ⊥ T,δ . From this it follows that L T,δ has a weight space decomposition withrespect to H T,δ with the set of roots R T,δ with ( L T,δ ) = H T,δ . Thus the axioms (A1)and (A2) of an EALA hold for L T,δ . The remaining axioms of an EALA holds triviallyfor L T,δ . Since R T,d is an affine root system, L T,δ has nullity 1.Next, we consider the subalgebra L aT,δ . If σ ∈ R and α, α + σ ∈ R × , then from theinvariance of the form and one-dimensionality of non-isotropic root spaces it followsthat ([ L α + σ , L − α ] , [ L α − σ , L − α ]) = 0. From this, we conclude that the form restricted to L aT,δ is also non-degenerate. Now an argument similar to the first statement shows that L aT,δ is an extended affine Lie algebra of nullity 1, with root system R T,δ . Therefore by[2, Theorem 2.31], it remains to show that L aT,δ is tame. For this, let L aT,δ,c denote thecore of L aT,δ and x ∈ L aT δ with ( x, L aT,δ,c ) = { } . We must show x ∈ L aT,δ,c . Since theform on L is H ⋆ -graded, namely ( L α , L β ) = { } unless α + β = 0, we get ( x, L β ) = 0for all β ∈ R × , implying that x ∈ L ⊥ c ⊆ L c . So x ∈ L c ∩ L aT,δ , forcing x ∈ L aT,δ,c asrequired. (cid:3) Lemma 3.5.
Let R be a reduced EARS of type X . Let α, β ∈ R be a pair of non-isotropic roots in R such that α + β is also a non-isotropic root in R . Then the set R α,β := R ∩ ( R α ⊕ R β ) is an irreducible reduced finite subsystem of R . Proof.
We prove the lemma through proving four claims as follows:
Claim 1 : ¯ α, ¯ β are non-proportional in ¯ V . If not, then as R is reduced, the onlypossibilities are ¯ β = ± ¯ α . But then in this case, if ¯ β = − ¯ α then α + β has to beisotropic which is impossible by the assumption, and, if ¯ β = ¯ α then 2 ¯ α ∈ ¯ R which isagain impossible. Hence Claim 1 follows. Claim 2 : R α,β is finite. We know that ¯ R is finite. Therefore, if R α,β is not finitethen there exist ˙ γ ∈ ˙ R and an infinite sequence δ n of isotropic roots in R such that˙ γ + δ n ∈ R α,β for every n ∈ N . Then for each n , there exist k n , k ′ n ∈ R such that(3.3) ˙ γ + δ n = k n α + k ′ n β. Now we have α = ˙ α + δ and β = ˙ β + δ ′ where ˙ α, ˙ β ∈ ˙ R are non-proportional and δ, δ ′ are isotropic. By (3.3) we have˙ γ + δ n = k n ˙ α + k ′ n ˙ β + k n δ + k ′ n δ ′ , for all n ∈ N . Since by Claim 1, ˙ α, ˙ β are non-proportional, this gives k n = k m and k ′ n = k ′ m for all n, m ∈ N . Thus δ n = δ m for all n, m ∈ N which is a contradiction. Claim 3 : R α,β is irreducible. It is enough to show that ¯ R α,β is irreducible. But thisis immediate as ¯ α, ¯ β are non-proportional and ¯ α + ¯ β is a root. Claim 4 : R α,β is an irreducible finite root system in V α,β := R α ⊕ R β . By Claims2,3, R is finite and irreducible. Moreover, the form restricted to V α,β is positive. Nowsince the subgroup W α,β of the Weyl group of R generated by the reflections associatedto α, β clearly preserves R α,β , and ( γ, η ∨ ) ∈ Z for all γ, η ∈ R α,β \ { } , we get fromRemark 2.2 that the claim holds. Consequence 3.6.
In the situation of Lemma 3.5, we haveType( R α,β ) = Type( ˙ R ˙ α, ˙ β ) . ROUPS OF EXTENDED AFFINE LIE TYPE 7
Proof.
By definition, R α,β has the same type of ¯ R α,β = ¯ R ∩ ( Z ¯ α ⊕ Z ¯ β ) ∼ = ˙ R ˙ α, ˙ β . Definition 3.7.
A pair { α, β } of non-isotropic roots in R satisfying α + β ∈ R × iscalled a nilpotent pair. According to Lemma 3.5, in this case R α,β is an irreduciblefinite root system.Let { α β } be a nilpotent pair. By Lemmas 3.5 and 3.2, L R α,β is a finite-dimensionalsimple Lie algebra of rank 2. Let L α,β be the subalgebra of L generated by L ± α , L ± β .It follows from [4, Lemma 1.3] that there exists a base { γ , γ } of R α,β such that L γ i ⊆ L α,β , i = 1 ,
2. It now follows again from [4, Lemma 1.3] that L α,β contains allroot spaces L γ , γ ∈ R α,β \ { } and so L α,β = L R α,β . We denote the nilpotent part ofthis Lie algebra obtained from positive roots by l α,β . Note that the above argumentshows that for each nilpotent pair { α, β } , we have L α,β = L α ′ ,β ′ where { α ′ , β ′ } is abase of R α ′ ,β ′ = R α,β . Proposition 3.8.
Let { α, β } be a nilpotent pair in R and δ = δ α + δ β . Then δ ∈ R ,moreover if δ = 0 , then R α,β,δ := R R α,β ,δ = R ( R α,β , S α,β , L α,β ) where S α,β := Z δ and L α,β = Z δ if R α,β is simply-laced, Z δ if R α,β is of type B = C , Z δ if R α,β is of type G , (see (2.1) for the notation R ( R α,β , S α,β , L α,β ) ). Proof.
With the notation of (2.1) let R = R ( ˙ R, S, L ). Put S ′ := S if α is short in R ,and put S ′ := L if α is long in R . Since α + β ∈ R , we have δ ∈ R . If δ = 0, we havefrom Proposition 3.3 that R α,β,δ is an irreducible affine root system. To complete theproof, we proceed in a few steps according to the type of R α,β . Suppose first that R α,β is simply laced. As α, β, α + β are in R α,β , we have R α,β = {± α, ± β, ± ( α + β ) } . Nowfor any n ∈ Z , we have ± α + nδ = ± ˙ α + ( n + 1) δ α + nδ β ∈ ± ˙ α + S ′ + 2 S ′ ⊆ ± ˙ α + S ′ ⊆ R. This shows that ± α + Z δ = ± α + S α,β ⊆ R. Similarly, we get ± β + S α,β ⊆ R . Also wehave ± ( α + β ) + S α,β = ± ( ˙ α + ˙ β ) + Z δ ⊆ ± ( ˙ α + ˙ β ) + S ′ ⊆ R , as required.Next, suppose that R α,β is not simply laced. As discussed in the paragraph precedingthe lemma, we may assume that { α, β } is a base of R α,β . Without loose of generality,assume that α is short and β is long. If we are in type B , then ±{ α, α + β } and ±{ β, α + β } are the sets of short and long roots of R α,β , respectively. Then we have ± α + S α,β = ± ( ˙ α + δ α ) + Z ( δ α + δ β ) ⊆ ± ˙ α + S + 2 S ⊆ ± ˙ α + S ⊆ R, ± ( α + β ) + S α,β = ± ( ˙ α + ˙ β + δ ) + Z δ ⊆ ± ( ˙ α + ˙ β ) + S ⊆ R, ± β + L α,β = ± ( ˙ β + δ β ) + 2 Z δ ⊆ ± ˙ β + L + 2 L ⊆ ± ˙ β + L ⊆ R, ± (2 α + β ) + L α,β = ± (2 ˙ α + ˙ β + 2 δ α + δ β ) + 2 Z δ ⊆ ± (2 ˙ α + ˙ β ) + 2 S + L + 2 S ⊆ ± (2 ˙ α + ˙ β ) + L ⊆ R. Finally, we consider the case in which R α,β is of type G . In this case ±{ α, α + β, β +2 α } and ±{ β +3 α, β +3 α, β +3 α } are the sets of short and long roots of R α,β , respectively.Now using an argument similar to the case B , using the facts that L + 3 S ⊆ L , S + L ⊆ S and S + 2 S ⊆ S , we are done. (cid:3) SAEID AZAM, AMIR FARAHMAND PARSA
Lemma 3.9.
Notations as in Lemma 3.5 and Proposition 3.8. Let W be the Weylgroup associated to a reduced EARS R . Then for any nilpotent pair of roots { α, β } andany w ∈ W we have (3.4) R α,β ∼ = wR α,β = R wα,wβ , and (3.5) R α,β,δ ∼ = wR α,β,δ = R wα,wβ,δ . Proof.
It follows from the fact that the form is invariant under the action of the Weylgroup, in particular W preserves root lengths. (cid:3) Groups of Extended Affine Lie Type
Groups Associated to EARSs.
We proceed with the same notations and as-sumptions as in the previous two sections. In particular, we assume ( L , H , ( · , · )) is atame EALA with root system R , R is reduced of type ˙ R ⊂ R and Λ ⊆ H ⋆ is the latticegenerated by isotropic roots.Let C σ be the associative C -algebra of the crossed product of Λ and C with respectto a 2-cocycle σ (see [19]). Note that the definition of C σ depends on the nullity of L (or equivalently the nullity of R ), and the chosen 2-cocycle σ . We call C σ the coordinatering associated to the L (or R ). The algebra C σ is known in the literature with differentnames such as Λ-torus, twisted group algebra, or quantum torus (see [9, § C σ has a basis { c λ | λ ∈ Λ } , and that themultiplication on C σ is determined by c λ c τ = σ ( λ, τ ) c λ + τ , ( λ, τ ∈ Λ) . When R is not of type A n (for some n ≥ σ is commutative.Note that the above condition on σ is not a major restriction on the simply laced casesby [24, Theorem 40.22(v&iv)] when we are mainly interested in group structures.For a pair of roots α, β ∈ R define R + α,β := ( N α + N β ) ∩ R . When R is a finite rootsystem, there exists an ordering, denoted by < , on the set of positive roots R + suchthat if α < β then β − α ∈ R + . On such sets of positive roots we always consider thisordering and call it the canonical order . In what follows, the term ( x, y ) := xyx − y − denotes the group commutator for two elements x and y of a group. Definition 4.1.
Let ˙ R be a finite reduced root system. Let A be an associative unitalring and U ( A ) the group of units of A . The group generated by ˆ x ˙ α ( a ) for all ˙ α ∈ ˙ R × and a ∈ A subject to the following relations is called the Steinberg group of type ˙ R over A and is denoted by St ˙ R ( A ): StF1: ˆ x ˙ α ( a ) is additive in a . StF2:
If rank( ˙ R ) ≥ α, ˙ β ∈ ˙ R , if ˙ α + ˙ β ∈ ˙ R × , then(4.1) (ˆ x ˙ α ( a ) , ˆ x ˙ β ( b )) = Y i ˙ α + j ˙ β ∈ ˙ R +˙ α, ˙ β ˆ x i ˙ α + j ˙ β ( c ij a i b j ) , ROUPS OF EXTENDED AFFINE LIE TYPE 9 with the canonical order on the finite set ˙ R + α,β , where c ij ’s in (4.1) are chosenas in [22, Lemma 15] corresponding to i ˙ α + j ˙ β ∈ ˙ R . StF2 ′ : If rank( ˙ R ) = 1, then(4.2) ˆ n ˙ α ( a )ˆ x ˙ α ( b )ˆ n ˙ α ( a ) − = ˆ x − ˙ α ( − a − b ) , for a ∈ U ( A ) and b ∈ A , where(4.3) ˆ n ˙ α ( a ) := ˆ x ˙ α ( a )ˆ x − ˙ α ( − a − )ˆ x ˙ α ( a ) . Let ˆ N and ˆ T denote the subgroups of St ˙ R ( A ) generated by ˆ n ˙ α ( a ) and ˆ h ˙ α ( a ) :=ˆ n ˙ α ( a )ˆ n ˙ α (1) − for a ∈ U ( A ), ˙ α ∈ ˙ R × . I.e,(4.4) ˆ N := h ˆ n ˙ α ( a ) | ˙ α ∈ ˙ R × , a ∈ U ( A ) i . and(4.5) ˆ T := h ˆ h ˙ α ( a ) | ˙ α ∈ ˙ R × , a ∈ U ( A ) i ⊆ ˆ N .
It immediately follows from (4.3) that(4.6) ˆ n ˙ α ( a ) − = ˆ n ˙ α ( − a )for a ∈ U ( A ), ˙ α ∈ ˙ R × . Lemma 4.2.
In the above setting, if A is commutative domain with characteristiczero, then the following relations in St ˙ R ( A ) hold: StF3: ˆ n ˙ α ( a )ˆ x ˙ β ( b )ˆ n ˙ α ( a ) − = ˆ x w ˙ α ˙ β ( c ( ˙ α, ˙ β ) a −h ˙ β, ˙ α i b ) , StF4: ˆ n ˙ α ( a )ˆ n ˙ β ( b )ˆ n ˙ α ( a ) − = ˆ n w ˙ α ˙ β ( c ( ˙ α, ˙ β ) a −h ˙ β, ˙ α i b ) , StF5: ˆ n ˙ α ( a )ˆ h ˙ β ( b )ˆ n ˙ α ( a ) − = ˆ h w ˙ α ˙ β ( c ( ˙ α, ˙ β ) a −h ˙ β, ˙ α i b )ˆ h w ˙ α ˙ β ( c ( ˙ α, ˙ β ) a −h ˙ β, ˙ α i ) − , StF6: ˆ n ˙ α ( a ) = ˆ n − ˙ α ( c ( ˙ α, ˙ β ) a − ) ,where c ( ˙ α, ˙ β ) = ± only depends on ˙ α and ˙ β and satisfies c ( ˙ α, ˙ β ) = c ( ˙ α, − ˙ β ) . Hence ˆ T is a normal subgroup of ˆ N . Proof.
Let ( A ) denote the field of fractions of A . The above relations hold for St ˙ R (( A ))by [22, Lemma 37]. By the functoriality of St ˙ R ( − ) we have St ˙ R ( A ) ֒ → St ˙ R (( A )). Hencethe lemma. (cid:3) Definition 4.3.
Let R be a reduced EARS of type ˙ R and σ a 2-cocycle on the latticeΛ = h R i . We define the Steinberg group St R, ˙ R,σ ( C ) associated to R over C to be thegroup generated by symbols x α ( t ) for each α ∈ R × and t ∈ C subject to the followingrelations: St1: x α ( t ) is additive in t . St2:
If rank( R ) ≥ α = ˙ α + δ α , β = ˙ β + δ β ∈ R × ⊆ ˙ R + Λ, if α + β ∈ R × (i.e., { α, β } is a nilpotent pair) then(4.7) ( x α ( s ) , x β ( t )) = Y iα + jβ ∈ R + α,β x iα + jβ ( c ij σ iδ α ,jδ β s i t j ) , with the canonical order on the finite set R + α,β , where c ij ’s in (4.7) are chosenas in [22, Lemma 15] corresponding to i ˙ α + j ˙ β ∈ ˙ R and(4.8) σ iδ α ,jδ β := σ ( iδ α , jδ β ) i − Y k =1 σ ( kδ α , δ α ) j − Y k =1 σ ( δ β , kδ β ) . St2 ′ : If rank( R ) = 1 then(4.9) n α ( t ) x α ( u ) n α ( t ) − = x − α ( − σ ( δ α , − δ α ) − t − u ) , for t ∈ C ∗ := C \{ } and u ∈ C , where(4.10) n α ( t ) := x α ( t ) x − α ( − σ ( δ α , − δ α ) − t − ) x α ( t ) . Consequence 4.4.
In the above setting, if α + β R × then ( x α ( t ) , x β ( s )) = 1. Proof. If α + β R × then R + α,β = ∅ by Definition 3.7. Hence ( x α ( t ) , x β ( s )) = 1 by St2 . (cid:3) Remark 4.5.
When R is of finite or affine type then the 2-cocycle σ ≡ § Remark 4.6.
When there exists no copy of G in R × , then the product in (4.7)does not depend on the chosen order since in such cases the right hand side of (4.7)has either one term ( R α,β = A ) or two terms ( R α,β = B ). And in the latter case, byConsequence 4.4, the two terms commute. In general, by [22, Chapter 3 Lemma 18] theproduct does not depend on any chosen order for the root set R + α,β up to isomorphism.For a reduced extended affine root system R = R ( ˙ R, S, L ), we set ˜ R := R ( ˙ R, h S i , h L i ).It is shown in [6, Section 2] that ˜ R is also an EARS with the same type, rank, nullityand twist number as R . We call ˜ R , the Λ- covering of R . Proposition 4.7.
Let R be a reduced EARS of type ˙ R . Then there exists an epimor-phism χ under which St ˙ R ( C σ ) is mapped onto St R, ˙ R,σ ( C ) . Furthermore, let κ χ be thekernel of χ . Then the following short exact sequence is right-split (4.11) κ χ → St ˙ R ( C σ ) χ → St R, ˙ R,σ ( C ) . Hence, (4.12) St ˙ R ( C σ ) ∼ = κ χ ⋊ St R, ˙ R,σ ( C ) . In particular, St ˙ R ( C σ ) is isomorphic to St ˜ R, ˙ R,σ ( C ) . Proof.
Define(4.13) St ˙ R ( C σ ) χ → St R, ˙ R,σ ( C ) , where(4.14) χ ( x ˙ α (Σ δ ∈ Λ t δ c δ )) := Y δ ∈ Λ x ˙ α + δ ( t δ ) . ROUPS OF EXTENDED AFFINE LIE TYPE 11
Note that (4.14) makes sense since only for finitely many δ ∈ Λ, the scalars t δ arenon-zero. Note that by convention if δ ∈ Λ and ˙ α + δ R × we define x ˙ α + δ ( t δ ) = 1. Wealso note that the right hand side of (4.14) does not depend on the order of its termsby Consequence 4.4 and that R is reduced.1) When rank( R ) ≥
2, by a similar argument as in [16, Proposition 3.2.11] theassignment χ defines a group homomorphism.2) Let rank( R ) = 1. It is enough to show that(4.15) χ ( n ˙ α ( tc δ )) χ ( x ˙ α ( rc δ )) χ ( n ˙ α ( − tc δ )) = χ ( x − ˙ α ( σ ( δ, − δ ) − t − rc − δ )) , for δ ∈ Λ, ˙ α ∈ ˙ R , t ∈ C ∗ and r ∈ C . But by the definition of χ , for any non-isotropicroot α = ˙ α + δ ∈ R × and any t ∈ C ∗ , the element ˆ n α ( t ) ∈ St ˙ R, Λ ,σ ( k ) correspondsto the element n ˙ α ( c δ t ) ∈ St ˙ R ( C σ ). Hence the proposition follows in rank one by thedefinition of χ (see (4.14)) and St2 ′ in Definition 4.3.In either case, χ is onto as for α = ˙ α + δ α ∈ R × and t ∈ k ∗ , χ ( x ˙ α ( tc δ α )) = x α ( t ).Furthermore, in both cases, similar to the way χ is defined, its right-inverse χ − r canbe defined by the following assignment:(4.16) χ − r ( x ˙ α + δ ( t )) = x ˙ α ( tc δ ) . For ˜ R , the map χ − r becomes onto, in particular χ for ˜ R is an isomorphism. (cid:3) Remark 4.8.
Let R = R ( ˙ R, S, L ) = R ( ˙ R ′ , S ′ , L ′ ) be two descriptions of R in the form(2.2). Let ψ be an isomorphism of R with ψ ( ˙ R ) = ˙ R ′ . The isomorphism ψ | Λ : Λ → Λinduces a 2-cocycle σ ′ := ψ ∗ σ . Then we have the commutative diagram(4.17) St ˙ R ( C σ ) χ (cid:15) (cid:15) ˆ ψ / / St ˙ R ′ ( C σ ′ ) χ ′ (cid:15) (cid:15) St R, ˙ R,σ ( C ) ˆ ψ / / St R, ˙ R ′ ,σ ′ ( C )where the upper and lower ˆ ψ ’s are the isomorphisms induced by ψ , namely x ˙ α ( c δ ) x ˙ α ′ ( c δ ′ ) and x α ( t ) x α ′ ( t ) respectively, where ˙ α ′ = ψ ( ˙ α ), δ ′ = ψ ( δ ) and α ′ = ψ ( α ),for ˙ α ∈ ˙ R , δ ∈ Λ and α ∈ R .We call the Steinberg group associated to ˜ R , the universal Steinberg group associatedto R .Let N denote the subgroup of St R, ˙ R,σ ( C ) generated by all n α ( t ) where α ∈ R × and t ∈ C ∗ . I.e.,(4.18) N := h n α ( t ) | α ∈ R × , t ∈ C ∗ i . Define(4.19) h α ( t ) := n α ( t ) n α (1) − for t ∈ C ∗ , (4.20) T := h h α ( t ) | α ∈ R × , t ∈ C ∗ i ⊂ N. Lemma 4.9.
In the above notations, we have T E N . Proof.
When ˙ R is of type A n (for some n ≥ σ is commutative and hence C σ constitutes acommutative (unital) domain with characteristic zero. Now by Lemma 4.2, ˆ T is normalin ˆ N ⊂ St ˙ R ( C σ ). Under the epimorphism χ in Proposition 4.7, N and T are the imagesof ˆ N and ˆ T respectively. The proposition follows. (cid:3) Definition 4.10.
In Definition 4.3 if we add the following relations, the resulting group G R, ˙ R,σ ( C ) is called the extended affine Kac–Moody group associated to ( R, ˙ R, σ ): Tor:
For every α ∈ R × , h α ( t ) is multiplicative in t ∈ C ∗ . Remark 4.11.
Note that since χ in Proposition 4.7 maps ˆ h ˙ α ( tc δ ) to h ˙ α + δ ( t ), it caneasily be checked that χ preserves the multiplicativity relation Tor in Definition 4.10.Let ˜ R be the Λ-covering of R . Then(4.21) G ˙ R ( C σ ) χ ∼ = G ˜ R, ˙ R,σ ( C ) , where G ˙ R is the universal Chevalley group scheme.4.2. Groups Associated to EALAs.
Here we construct certain groups associatedto EALAs by means of certain representations of Lie algebras; this method is knownas the “Kac-Peterson method” (see for example [15]), to achieve this we follow closely[16].Let L be an EALA with a reduced EARS R = R ( ˙ R, S, L ). Let G nil ( G int ) be thegroup associated to L with respect to nilpotent (integrable) representations of L . Sim-ilarly, let G nil,c ( G int,c ) be the group associated to L c with respect to nilpotent (inte-grable) representations of L c where L c is the core of L (see [16, Section 3.2] and [17,Section 6.1] for details). To emphesize on the underlined filed we sometimes write G ( C )instead of G , for G ∈ { G nil , G int , . . . } .We generalize [16, Proposition 3.2.29] for any EALA with a reduced EARS. Recallour standing assumption that in the case ˙ R is not of type A n (for some n ∈ N ) wealways assume that the 2-cocycle σ is commutative. Proposition 4.12.
Let L be an EALA with a reduced EARS R = R ( ˙ R, S, L ) andthe coordinate ring C σ . Then for each G ∈ { G nil , G int , G nil,c , G int,c } there exist epi-morophisms Π G and π G such that the following diagram commutes. (4.22) St ˙ R ( C σ ) χ (cid:15) (cid:15) Π G / / G St R, ˙ R,σ ( C ) π G : : ✈✈✈✈✈✈✈✈✈✈ Proof.
When ˙ R = A n (for some n ≥
2) this is [16, Proposition 3.2.29]. Assume that˙ R is not of type ˙ R = A n (for any n ≥ α ∈ ˙ R × and δ ∈ Λ and any nilpotent representation ρ of L define(4.23) Π ρG nil (ˆ x ˙ α ( tc δ )) = ¯ ρ (exp( tc δ X ˙ α )) , ROUPS OF EXTENDED AFFINE LIE TYPE 13 and similarly for α = ˙ α + δ α ∈ R × ,(4.24) π ρG nil ( x α ( t )) = ¯ ρ (exp( tc δ α X ˙ α )) , where X ˙ α is chosen as in [22, Lemma 15] associated to ˙ α ∈ ˙ R × and ¯ ρ is the inducedrepresentation of G nil corresponding to ρ , and t ∈ k . If Π ρG nil and π ρG nil define homo-morphisms, then clearly by their definitions and the definition of χ in Proposition 4.7,the diagram (4.22) is commutative. To show that Π ρG nil and π ρG nil are homomorphisms,it suffices to check that the defining relations in St ˙ R ( C σ ) and St R, ˙ R,σ ( k ) hold for theimage of elements in G nil via Π ρG nil and π ρG nil respectively. It is clear that for suchelements (St1) in Definition 4.3 holds.1) Assume rank( ˙ R ) ≥
2. If α + β ∈ R × , then, by Lemma 3.5, R α,β is an irreduciblereduced finite root system and L α,β is a finite-dimensional simple Lie algebra. Let ( C σ )be the fraction field of C σ . By [22, Lemma 15], relation (St2) of Definition 4.3 holds inthe Lie algebra L α,β over ( C σ ). Hence (St2) also holds for coefficients from C σ ⊂ ( C σ ).Note that in the case of π ρG , this argument is still valid since for every pair of nilpotentroots α and β , R α,b is a finite root system by Lemma 3.5 therefore all of its non-zeroroots are non-isotropic and hence none of them is in the kernel of π ρG .2) Assume rank( ˙ R ) = 1. For α = ˙ α + δ α ∈ R × by (4.23) we have(4.25) Π ρG nil (ˆ x ˙ α ( tc δ α )) = ¯ ρ (cid:18)(cid:18) tc δ α (cid:19)(cid:19) , for t ∈ C and(4.26) Π ρG nil (ˆ n ˙ α ( tc δ α )) = ¯ ρ (cid:18)(cid:18) tc δ α − σ ( δ α , − δ α ) − t − c − δ α (cid:19)(cid:19) , for t ∈ C ∗ . Now, it is a simple matrix calculation to see Definition 4.3(St2 ′ ) holds as well.The same argument is valid for π ρG nil . Moreover, elements of the form ¯ ρ (exp( tc δ α X ˙ α ))generate G nil . Hence Π ρG nil (similarly π ρG nil ) induces an epimorphism. Since the aboveargument is independent of the choice of representation and it is valid also for any G ∈ { G nil , G int , G nil,c , G int,c } , the proposition follows. (cid:3) For a non-isotropic root α ∈ R × , let G α ∗ ( C ) be the group in G ∗ ( C ) generated by U ± α ( C ) := h Π G ∗ (ˆ x ± ˙ α ( tc ± δ )) | t ∈ C i (Π G ∗ as in Proposition 4.12) where G ∗ ( C ) ∈ { G nil ( C ) , G int ( C ) , G nil,c ( C ) , G int,c ( C ) } . For such a non-isotropic root α we also know that sl α := L α + [ L α , L − α ] + L − α is acopy of sl with a standard basis { e α , h α , f α } .The following is a generalization of [16, Proposition 3.2.30]. Proposition 4.13.
Let L be an EALA with a reduced EARS R over C . Let ( ρ, M ) bean integrable L -module. Then the following hold. (i) For every non-isotropic root α ∈ R × there exists an epimorphism (4.27) ϕ αρ : SL ( C ) → ¯ ρ ( G α ∗ ( C )) , such that (4.28) ϕ αρ (cid:18)(cid:18) t (cid:19)(cid:19) = ¯ ρ (exp te α ) , ϕ αρ (cid:18)(cid:18) t (cid:19)(cid:19) = ¯ ρ (exp tf α ) , for all t ∈ C , where (¯ ρ, M ) is the representation induced by ( ρ, M ) of G ∗ ( C ) ∈{ G int ( C ) , G int,c ( C ) } . (ii) Assume either rank ( R ) ≥ or the nullity of L is 1. Then there exists a uniqueisomorphism (4.29) ϕ α : SL ( C ) → G α ∗ ( C ) , such that for all t ∈ C (4.30) ϕ α (cid:18)(cid:18) t (cid:19)(cid:19) = exp te α , ϕ α (cid:18)(cid:18) t (cid:19)(cid:19) = exp tf α , and (4.31) ¯ ρ ◦ ϕ α = ϕ αρ , where G ∗ ( C ) ∈ { G int ( C ) , G int,c ( C ) } . Proof. (i) Follows from the proof of [17, Proposition 6.1.7(i)].(ii) First assume that rank( R ) ≥
2. In this case, let α = ˙ α + δ α ∈ ˙ R + R . Sincerank( R ) ≥
2, there exists ˙ β ∈ ˙ R such that ˙ β = ± ˙ α and ˙ α + ˙ β ∈ ˙ R . Moreover, from theclassification of finite root systems one can see that in the non-simply laced cases, ˙ β can be chosen such that ˙ α + ˙ β ∈ ˙ R sh . Therefore, in all cases α + ˙ β = ˙ α + ˙ β + δ α ∈ R × and so { α, ˙ β } forms a nilpotent pair in R . Hence, (ii) follows from Lemma 3.4 and [17,Proposition 6.1.7(ii)]. When the nullity is equal to one, L is an affine Kac–Moody Liealgebra. Therefore, (ii) directly follows from [17, Proposition 6.1.7(ii)]. (cid:3) For a non-isotropic root α ∈ R × , let ¯ h α ( t ) := π G int ( h α ( t )) for t ∈ C ∗ where h α ( t ))is as in (4.19) and π G int as in Proposition 4.12. Then we have the following corollary. Corollary 4.14.
For any non-isotropic root α ∈ R × we have (4.32) ¯ h α ( st ) = ¯ h α ( s )¯ h α ( t ) . Proof.
When either rank( R ) ≥ R ) = 1 and the nullity of L is greateror equal to 2, then the corollary follows from (4.25) and (4.26) in Proposition 4.12 (moreprecisely, adjusted formula for π G int , see (4.24)). (cid:3) Consequence 4.15.
In the above setting, for G int ( C ) there exists an epimorphisminduced from π G in Proposition 4.12, namely:(4.33) ¯ π G int : G R, ˙ R,σ ( C ) → G int ( C ) . Proof.
It immediately follows from Corollary 4.14 that π G preserves Tor in Defini-tion 4.10, hence the result. (cid:3)
Similar to ¯ h α ( t ), let ¯ x α ( t ) := π G int ( x α ( t )) for t ∈ C and ¯ n α ( t ) := π G int ( n α ( t )) for t ∈ C ∗ . Also we define ¯ N and ¯ T correspondingly. ROUPS OF EXTENDED AFFINE LIE TYPE 15
Consequence 4.16. ¯ T E ¯ N . Proof.
Follows directly from Lemma 4.9 and Proposition 4.12. (cid:3)
Corollary 4.17.
With notations as above we have: (i)
When rank ( R ) ≥ , then ¯ n α ( t )¯ x λ ( u )¯ n α ( t ) − = ¯ x w α ( λ ) ( t −h λ,α i u ) , where λ = ± α , t ∈ C ∗ and u ∈ C . (ii) When rank ( R ) = 1 , then ¯ n α ( t )¯ x λ ( u )¯ n α ( t ) − = ¯ x − λ ( − σ ( δ α , − δ α ) − t −h λ,α i u ) , where α = ˙ α + δ α , λ = ± α , t ∈ C ∗ and u ∈ C . (iii) In general, if σ is commutative and α = ˙ α + δ α and β = ˙ β + δ β are both in R × , then ¯ n α ( t )¯ x β ( u )¯ n α ( t ) − = ¯ x w α ( β ) ( λ α,β t −h β,α i u ) , where λ α,β = c ( ˙ α, ˙ β ) · σ ( − h β, α i δ α , δ β ) c ( ˙ α, ˙ β ) |−h β,α i− c ( ˙ α, ˙ β ) | Y k = | c ( ˙ α, ˙ β ) | σ ( kδ α , δ α ) c ( ˙ α, ˙ β ) and c ( ˙ α, ˙ β ) = ± only depends on ˙ α and ˙ β . Proof. (i) and (ii): Similar to the proof of Corollary 4.14.(iii): First note that by StF3 in Lemma 4.2, we haveˆ n ˙ α ( a )ˆ x ˙ β ( b )ˆ n ˙ α ( a ) − = ˆ x w ˙ α ˙ β ( c ( ˙ α, ˙ β ) a −h ˙ β, ˙ α i b ) , henceˆ n ˙ α ( tc δ α )ˆ x ˙ β ( uc δ β )ˆ n ˙ α ( tδ α ) − =ˆ x w ˙ α ˙ β (cid:0) c ( ˙ α, ˙ β ) t −h β,α i uσ ( − h β, α i δ α , δ β ) c ( ˙ α, ˙ β ) Q k σ ( kδ α , δ α ) c ( ˙ α, ˙ β ) c −h β,α i δ α + δ β (cid:1) , where k ranges from | c ( ˙ α, ˙ β ) | to | − h β, α i − c ( ˙ α, ˙ β ) | . But this equation is mapped tothe following via Π ρG as (4.23) in Proposition 4.12:¯ n α ( t )¯ x β ( u )¯ n α ( t ) − = ¯ x w α ( β ) ( λ α,β t −h α,β i u ) . (cid:3) Remark 4.18.
Note that in Corollary 4.17 the difference between (i) and (ii) is due tothe fact that in (i) any non-isotropic root can be embedded into an affine root systemof rank two (see Proposition 3.8) and on the corresponding affine Lie subalgebra wehave σ ≡ R ) = 1. Lemma 4.19.
Let L be an EALA over C . Then every integrable (respectively, nilpo-tent) L c -module can be extended to an integrable (respectively, nilpotent) L -module. Proof.
It immediately follows from [16, Lemma 3.2.32] and the explanation after itsproof on Page 130. (cid:3)
Let K nil (respectively, K int ) be the intersection of the kernel of all nilpotent (respec-tively, integrable) representations of an EALA L over C . Similarly, we define K nil,c and K int,c for the core L c of L . Proposition 4.20.
Let L be an EALA with a reduced EARS R . Then the followinghold. (i) K nil ⊆ K nil,c . (ii) K int ⊆ K int,c .Therefore, there exist natural epimorphisms as follows (4.34) Ψ nil : G nil → G nil,c , Ψ int : G int → G int,c . Proof.
Follows from Lemma 4.19 and the proof of [16, Proposition 3.2.38]. (cid:3)
Define epimorphisms(4.35) Ψ nil ,c := Ψ nil ◦ π G nil : St R, ˙ R,σ ( C ) → G nil,c ( C ) , and(4.36) Ψ int ,c := Ψ int ◦ π G int : St R, ˙ R,σ ( C ) → G int,c ( C ) , where π G nil and π G int are as in Proposition 4.12. Consequence 4.21.
Let L be an EALA with a reduced EARS R = R ( ˙ R, S, L ) andlet C σ be a coordinate ring. Then there exists an epimorphism from G R, ˙ R,σ ( C ) onto G int,c , namely(4.37) ¯ π G int ,c : G R, ˙ R,σ ( C ) → G int ,c ( C ) . Proof.
Follows from Consequence 4.15 and Proposition 4.20. (cid:3)
By the above adjustment, the next proposition is a generalization of [16, Proposition3.2.41]. Its proof is also similar to the proof of [16, Proposition 3.2.41].
Proposition 4.22.
In the above setting we have G nil ( C ) ∼ = G nil ,c ( C ) ∼ = G int ( C ) ∼ = G int,c ( C ) . In the light of Proposition 4.22, from now on we denote any of the above isomorphicgroups simply by ¯ G . Also we denote the group corresponding to the adjoint representa-tion by Ad( G ). Similarly, Ad( N ) and Ad( T ) denote the groups corresponding to ¯ N and¯ T respectively. It is evident from Consequence 4.16 that Ad( T ) E Ad( N ). Moreover,elements θ α ( t ) := Ad(¯ n α ( t )) for α ∈ R × and t ∈ C are of the following form:(4.38) θ α ( t ) := exp ad te α exp ad − t − f α exp ad te α , where sl α := L α +[ L α , L − α ]+ L − α is a copy of sl with a standard basis { e α , h α , f α } . Fora non-isotropic root α ∈ R × and an isotropic root σ = P νi =1 m i σ i ∈ R (null( R ) = ν )let c ( α,σ ) be as in (2.3). ROUPS OF EXTENDED AFFINE LIE TYPE 17
Theorem 4.23.
Let L be an EALA over C with a reduced EARS R = R ( ˙ R, S, L ) . Let W be the Weyl group associated to L . Then (4.39) W Φ ∼ = Ad ( N ) / Ad ( T ) , where Φ sends w α to Ad ( T ) Ad (¯ n α ( t )) for all α ∈ R × and t ∈ C ∗ . Proof.
When ˙ R = A n (for n ≥
2) this is [16, Proposition 3.3.1]. We now consider allother possible cases. Similar to the proof of [16, Proposition 3.3.1], it is clear that Φdoes not depend on the choice of t ∈ C ∗ . First we show that Φ defines an epimorphism.For this, note that by Theorem 2.3, in view of Proposition 4.12, it is enough to showthat elements of the form ˜ w α := Ad( T )Ad(¯ n α ( t )) satisfy the generalized presentationby conjugation relations for W . Recall from Proposition 4.12 that elements ˆ n ˙ α ( c δ α ) in St ˙ R ( C σ ) correspond to Ad(¯ n α (1)) where α = ˙ α + δ α ∈ R × . Since both Ad( T )Ad(¯ n α (1))and Ad( T )Ad(¯ n α ( − w α by (4.6), it is clear that ˜ w α = 1. Hencerelation (i) in Theorem 2.3 holds.Let Rank( R ) ≥
2. In this case, for any β = ˙ β + δ β , by Lemma 4.2(StF4), we have(4.40) ˆ n ˙ α ( c δ α )ˆ n ˙ β ( c δ β )ˆ n ˙ α ( c δ α ) − = ˆ n w ˙ α ˙ β ( κ ( α, β ) c − ( ˙ β, ˙ α ) δ α c δ β ) χ = n w α β ( κ ( α, β )) , for some κ ( α, β ) ∈ C ∗ . Therefore, ˜ w α ˜ w β ˜ w α = ˜ w w α ( β ) and hence relation (ii) in Theorem2.3 also holds. Moreover, when Rank( R ) = 1 by (4.25) and (4.26) and a straightforwardmatrix calculation one can easily see that relation (ii) also holds in this case.Now consider c ( α,δ ) as in (2.3), we have:(4.41) ˜ c ( α,δ ) := (ˆ n ˙ α ( c δ α + δ )ˆ n ˙ α ( c δ α ))(ˆ n ˙ α ( c δ α )ˆ n ˙ α ( c δ α + δ )) m · · · (ˆ n ˙ α ( c δ α )ˆ n ˙ α ( c δ α + δ ν )) m ν , which is equal to(4.42)(ˆ h ˙ α ( c δ α + δ )ˆ h ˙ α ( − c δ ) − )(ˆ h ˙ α ( c δ )ˆ h ˙ α ( − c δ α + δ ) − ) m · · · (ˆ h ˙ α ( c δ ν )ˆ h ˙ α ( − c δ α + δ ν ) − ) m ν , whose image under Π G ◦ Ad (Π G as in Proposition 4.12) is in Ad( T ), so is the corre-sponding element to Q np =1 ˜ c ǫ p ( α p ,δ p ) for any collection { ( ǫ p , α p , δ p ) } np =1 . Hence relation(iii) in Theorem 2.3 holds as well.At last, by [1, (1.26)] we know that the restriction of the action of θ α ( t ) (for all α ∈ R × and t ∈ C ∗ ) on the Cartan subalgebra H of L is the same as the action of theWeyl group element w α , so the theorem follows. (cid:3) Acknowledgements.
The second-named author would like to thank Ralf K¨ohl formany useful discussions on the preliminary version of this article during a visit toGießen university.
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