Morita equivalences for cyclotomic Hecke algebras of type B and D
aa r X i v : . [ m a t h . R T ] N ov Morita equivalences for cyclotomic Hecke algebras of type Band DÉquivalences de Morita pour les algèbres de Hecke cyclotomiquesde type B et D
Loïc
Poulain d’Andecy ∗† Salim
Rostam ‡ Abstract
We give a Morita equivalence theorem for so-called cyclotomic quotients of affine Heckealgebras of type B and D, in the spirit of a classical result of Dipper–Mathas in type A forAriki–Koike algebras. Consequently, the representation theory of affine Hecke algebras oftype B and D reduces to the study of their cyclotomic quotients with eigenvalues in a singleorbit under multiplication by q and inversion. The main step in the proof consists in adecomposition theorem for generalisations of quiver Hecke algebras that appeared recentlyin the study of affine Hecke algebras of type B and D. This theorem reduces the generalsituation of a disconnected quiver with involution to a simpler setting. To be able to treattypes B and D at the same time we unify the different definitions of quiver Hecke algebrafor type B that exist in the literature. Résumé
Nous énonçons un théorème d’équivalence de Morita pour les quotients cyclotomiquesdes algèbres de Hecke affines de type B et D, suivant un résultat classique de Dipper–Mathasen type A pour les algèbres d’Ariki–Koike. Ainsi, la théorie des représentations des algèbresde Hecke affines de type B et D se réduit à l’étude de leurs quotients cyclotomiques oùles valeurs propres sont dans une unique orbite pour la multiplication par q et l’inversion.La preuve consiste notamment en un théorème de décomposition pour des généralisationsd’algèbres de Hecke carquois introduites récemment dans l’étude des algèbres de Hecke affinesde type B et D, ramenant la situation générale d’un carquois non connexe avec involution àun cadre plus simple. Pour traiter simultanément les deux types, nous unifions les différentesdéfinitions d’algèbres de Hecke carquois pour le type B déjà existantes. Introduction
Cyclotomic quotients of the affine Hecke algebra of type A, also known as Ariki–Koikealgebras, have been extensively studied since their introduction by Broué–Malle [5] andAriki–Koike [2]. Given a field K , a subset I ⊆ K × , an element q ∈ K × and a finitely-supported family Λ = (Λ i ) i ∈ I of non-negative integers, the Ariki–Koike algebra H Λ ( S n ) is ∗ The first author is supported by
Agence Nationale de la Recherche through the JCJC project ANR-18-CE40-0001. † Laboratoire de Mathématiques de Reims UMR 9008, Université de Reims Champagne-Ardenne, Moulin dela Housse BP 1039, 51100 Reims, France ‡ Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France efined by the generators g , . . . , g n − and the relations g i g j = g j g i , for all i, j ∈ { , . . . , n − } , | i − j | > ,g i g i +1 g i = g i +1 g i g i +1 , for all i ∈ { , . . . , n − } ,g g g g = g g g g , ( g i − q )( g i + q − ) = 0 , for all i ∈ { , . . . , n − } , Y i ∈ I ( g − i ) Λ i = 0 . We note that Ariki–Koike algebras are quotients, by the last relation, of affine Hecke algebrasof type A and that the study of their representations (for all choices of I and Λ) is equivalentto the study of finite-dimensional representations of affine Hecke algebras of type A.By an important theorem of Dipper–Mathas [8], we know that it suffices to study Ariki–Koike algebras when the set I is q -connected , that is, in a single q -orbit (and even, up toa scalar renormalisation of the generator g , when I ⊆ h q i ). More precisely, if I = ∐ dj =1 I ( j ) is the decomposition of I into q -connected sets then we have a Morita equivalence H Λ ( S n ) Morita ≃ M n ,...,n d ≥ n + ··· + n d = n d O j =1 H Λ ( j ) ( S n j ) , ( ♣ )where Λ ( j ) is the restriction of Λ to I ( j ) . (Note that the assumption in [8] is slightly strongerthan the one above, but in practice it is this condition of q -connected sets that is used.)Hence, this Morita equivalence allows to use results that are only known when the set I is q -connected, in particular, the celebrated Ariki’s categorification theorem [1] that computesthe decomposition numbers of Ariki–Koike algebras in terms of the canonical basis of acertain highest weight module over an affine quantum group.Another way to obtain this Morita equivalence was given by the second author [22, §3.4],using the theory of quiver Hecke algebras. This is a family of graded algebras that wasintroduced a few years ago independently by Khovanov–Lauda [16, 17] and Rouquier [23],in the context of categorification of quantum groups. If Γ is a quiver, we denote by R n (Γ)the associated quiver Hecke algebra (see §2.1). For a certain quiver Γ depending only on theorder of q , Brundan–Kleshchev [6] and independently Rouquier [23] proved that a certain“cyclotomic” quotient of R n (Γ) is isomorphic to an Ariki–Koike algebra. This result is nowa basic tool in the study of Ariki–Koike algebras and their degenerations, including thesymmetric group and the classical Hecke algebra of type A. For instance, as consequencesfirst the Ariki–Koike algebra inherits the grading of the cyclotomic quiver Hecke algebra,and second depends on q only through its order in K × . Now if Γ is of the form Γ = ∐ dj =1 Γ ( j ) where each Γ ( j ) is a full subquiver, it was shown in [21, §6] that we have a decomposition R n (Γ) ≃ M n ,...,n d ≥ n ··· + n d = n Mat( nn ,...,nd ) d O j =1 R n j (Γ ( j ) ) . ( ♠ )This isomorphism of algebras is compatible with cyclotomic quotients, and combining withthe previous isomorphism of Brundan–Kleshchev and Rouquier allows to recover the Moritaequivalence ( ♣ ). This Morita equivalence has been further generalised for the cyclotomicHecke algebras of type G ( r, p, n ) [11]. We indicate also the paper [12] where the Dipper–Mathas result is studied and derived again from the point of view of affine Hecke algebras,and where the question of a similar result for other affine Hecke algebras is evoked.The main point of this paper is to prove a similar decomposition theorem for somegeneralisations of quiver Hecke algebras and hence obtain an analogue of the Dipper–MathasMorita equivalence for cyclotomic quotients of affine Hecke algebras of type B and D. Suchgeneralisations of quiver Hecke algebras were introduced by Varagnolo and Vasserot [25] (for ype B) and together with Shan [24] (for type D), in the course of their proofs of conjecturesby Kashiwara–Enomoto [9] and Kashiwara–Miemietz [15]. These algebras play for certainsubcategories of representations of affine Hecke algebras of type B and D a similar role asquiver Hecke algebras for affine Hecke algebras of type A. Inspired by their results, thefirst author together with Walker [19, 20] obtained an isomorphism theorem à la Brundan–Kleshchev between cyclotomic quotients of affine Hecke algebras of type B and D and certaingeneralisations of cyclotomic quiver Hecke algebras.The first step of this paper is to provide a definition of these generalisations of quiverHecke algebras for the type B which encompasses all the slightly different versions previouslydefined. They are Z -graded algebras and they depend upon a quiver with an involution andcertain weight functions on the vertices. As for the type A case, that is, for usual quiver Heckealegbras, the algebra that we define admits a PBW basis and this is a key ingredient to provethe decomposition theorem when the underlying quiver has several connected components.The point of having defined a new algebra in Section 3 is that we can now use the mainresults of [19, 20] at the same time. We deduce our main theorem for type B, Theorem 6.8,that we state now. Write I ⊆ K × as I = ∐ dj =1 I ( j ) such that each I ( j ) is q -connected and stable by scalar inversion. As in the type A case, for Λ = (Λ i ) i ∈ I ∈ N ( I ) we denote by H Λ ( B n ) the quotient of the affine Hecke algebra of type B by the relation Y i ∈ I ( X − i ) Λ i = 0(see §6.1 for a precise definition). Theorem.
We have an (explicit) isomorphism H Λ ( B n ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) d O j =1 H Λ ( j ) ( B n j ) , in particular, we have a Morita equivalence H Λ ( B n ) Morita ≃ M n ,...,n d ≥ n + ··· + n d = n d O j =1 H Λ ( j ) ( B n j ) . We also deduce that a similar result holds for the cyclotomic quotient H Λ ( D n ) of theaffine Hecke algebra of type D. Some technicalities typical to the type D situation result in aformulation of the final result a bit more complicated than for type B in the Theorem above,since it involves in addition a semi-direct product by powers of a cyclic group of order 2 (seeTheorem 6.19).One motivation for considering cyclotomic quotients of affine Hecke algebras is that thestudy of (finite-dimensional) representations of the affine Hecke algebra is equivalent to thestudy of representations of all their cyclotomic quotients. As a consequence of our mainresults, we obtain that, for affine Hecke algebras of type B and D, this study reduces toconsidering the algebras H Λ ( B n ) and H Λ ( D n ) when the set I is q -connected and stable byscalar inversion (see Corollaries 6.9 and 6.20 for more details and a complete description ofthe finite number — up to four — of sets I to be considered). This generalises the classicalreduction for the affine Hecke algebras of type A (for which it is enough to consider I = q Z )induced by the Dipper–Mathas result. Organisation of the paper.
In Section 1, given an algebra A and a set of idempotentssatisfying certain properties we prove a general decomposition theorem expressing A in termsof a direct sum involving matrix algebras on idempotent truncations (Corollary 1.13).Let Γ be a (possible infinite) quiver with no 1-loops, let I be its vertex set and let α ⊆ S n be a finite union of S n -orbits. In Section 2 we recall the definition of the quiver Hecke algebra R α (Γ). We then review the proof, based on the general theorem from Section 1, of the ecomposition isomorphism of [21] when Γ has several connected components, generalisingit to the case where Γ is not necessarily finite (as it is assumed in [21]). In §2.2.4, given afinitely-supported family Λ of non-negative integers we define the cyclotomic quotient R Λ α (Γ)of R α (Γ) and give the corresponding isomorphism when Γ has several connected components.Then we assume that Γ is endowed with an involution θ and let β ⊆ I n be an orbit forthe action of the Weyl group B n of type B and rank n . We begin Section 3 by defining thealgebra V β (Γ , λ, γ ) depending in addition on λ ∈ N I and γ ∈ K I satisfying certain conditions.This algebra generalises the constructions of [25, 19, 20], see Remarks 3.16, 3.17 and 3.18respectively. The algebra V β (Γ , λ, γ ) is Z -graded, and we prove in §3.2 that it admits a PBWbasis, using a polynomial realisation (the calculations are postponed to Appendix A).Section 4 is the heart of the paper. We prove a decomposition theorem, similar to ( ♠ ),for the algebra V β (Γ , λ, γ ) when the quiver Γ is a disjoint union of θ -stable full subquiversΓ = ∐ dj =1 Γ ( j ) (Theorem 4.1). As in Section 2, we first use the results of Section 1 and thenprove that some idempotent truncation of V β (Γ , λ, γ ) can be expressed as a tensor producton smaller algebras involving the quivers Γ ( j ) . Note here a technical difficulty comparingwith the type A case: for n + · · · + n d = n , the group S n × · · · × S n d can be seen as aparabolic subgroup of S n for its standard Coxeter structure, but it is no more the case for B n × · · · × B n d ⊆ B n , although this is still a subgroup. We prove in §4.3 the cyclotomicanalogue of the decomposition theorem (Corollary 4.17).The shorter Section 5 is devoted to quiver Hecke algebras W β (Γ) for type D and theircyclotomic quotients W Λ β (Γ). Using a result of [20] that expresses W β (Γ) as the subalgebraof fixed-points of a certain involutive automorphism of V β (Γ , ,
0) (Proposition 5.17), wemanage to give a decomposition isomorphism for W β (Γ) and its cyclotomic quotient whenthe quiver Γ has several θ -stable full subquivers (Theorem 5.26).Finally, in Section 6 we introduce the affine Hecke algebras H ( B n ) of type B and H ( D n )of type D, together with their cyclotomic quotients H Λ ( B n ) and H Λ ( D n ). We then usethe analogues of Brundan–Kleshchev isomorphism theorem in types B and D from [19,20] to deduce from our disjoint quiver isomorphisms the announced Morita equivalences:Theorem 6.8 for type B and Theorem 6.19 for type D. Acknowledgements
The authors would like to thank Ruari Walker for many interestingdiscussions initiating this work. The second author would like to thank Ruslan Maksimaufor explaining a proof of Proposition 2.12. The authors are very grateful to an anonymousreferee for many useful suggestions.
The results in this section, or some versions of them, are probably known to specialists, butwe could not find them in this precise form in the literature. So we state them in the formwe need and provide complete proofs. The framework presented here encompasses severalcases of proved isomorphism theorems such as in [13, 21].Let A be a unitary algebra over a ring K . Let I be a complete (finite) set of orthogonalidempotents, that is:• for all e ∈ I we have e = e ;• for all e, e ′ ∈ I , if e = e ′ then ee ′ = e ′ e = 0;• we have 1 = P e ∈I e .For any e ∈ I , let φ e , ψ e ∈ A such that φ e ψ e e = e, (1.1a) eφ e ψ e = e. (1.1b) Remark . Such elements necessarily exist, for instance φ e = ψ e = e for any e ∈ I .However, obviously this will not lead to interesting results. emma 1.3. For any e ∈ I , the element ψ e eφ e is an idempotent.Proof. Using (1.1a), we have ( ψ e eφ e ) = ψ e e ( φ e ψ e e ) φ e = ψ e e φ e = ψ e eφ e , as desired.Denote by J the image of the map I −→ Ae ψ e eφ e and write I ǫ for the fibre of anyelement ǫ ∈ J . We have I ǫ = { e ∈ I : ψ e eφ e = ǫ } , and G ǫ ∈J I ǫ = I . By Lemma 1.3, the set J consists of idempotents, however it is a priori not related to I . Proposition 1.4.
For any ǫ ∈ J and any e ∈ I ǫ we have eφ e = φ e ǫ, (1.5a) ǫψ e = ψ e e. (1.5b) Proof.
We have ψ e eφ e = ǫ, (1.6)thus ( φ e ψ e e ) φ e = φ e ǫ . Using (1.1a) we obtain the first equality. We also obtain ψ e ( eφ e ψ e ) = ǫψ e from (1.6) thus by (1.1b) we obtain the second equality. Proposition 1.7.
For any ǫ ∈ J and any e ∈ I ǫ we have ψ e φ e ǫ = ǫ, (1.8a) ǫψ e φ e = ǫ. (1.8b) Proof.
By (1.5a) we have φ e ǫ = eφ e , thus ψ e φ e ǫ = ψ e eφ e , and we conclude that (1.8a) holds since ψ e eφ e = ǫ by definition of I ǫ . Similarly, by (1.5b)we have ǫψ e φ e = ψ e eφ e = ǫ, thus (1.8b) holds.If J is any finite set and B any K -algebra, we denote by Mat J ( B ) the K -algebra ofmatrices with rows and columns indexed by J with entries in B . Definition 1.9.
For any ǫ ∈ J , we define the idempotentˆ ǫ := X e ∈I ǫ e. Theorem 1.10.
Let ǫ ∈ J . We have the following isomorphism of K -algebras: ˆ ǫA ˆ ǫ ≃ Mat I ǫ ( ǫAǫ ) . roof. We first prove that for any e ′ , e ∈ I ǫ , the maps θ e ′ e : e ′ Ae −→ ǫAǫM e ′ e a ψ e ′ aφ e M e ′ e , and η e ′ e : ǫAǫM e ′ e −→ e ′ AeaM e ′ e φ e ′ aψ e , are well-defined and inverse isomorphism of K -modules. Here, we denoted by M e ′ e ∈ Mat I ǫ ( ǫAǫ ) the matrix whose unique non-zero coefficient, which is 1, is at row e ′ and col-umn e . The maps θ e ′ e and η e ′ e are well-defined by Proposition 1.4. Indeed, for any a ∈ e ′ Ae then a = e ′ ae and ψ e ′ aφ e = ( ψ e ′ e ′ ) a ( eφ e ) = ( ǫψ e ′ ) a ( φ e ǫ ) ∈ ǫAǫ, so θ e ′ e is well-defined, and for any a ∈ ǫAǫ then a = ǫaǫ and φ e ′ aψ e = ( φ e ′ ǫ ) a ( ǫψ e ) = ( e ′ φ e ′ ) a ( ψ e e ) ∈ e ′ Ae, so η e ′ e is well-defined. Now for any a ∈ e ′ Ae we have, using a = e ′ ae and (1.1), η e ′ e (cid:0) θ e ′ e ( a ) (cid:1) = η e ′ e ( ψ e ′ aφ e M e ′ e )= φ e ′ ( ψ e ′ aφ e ) ψ e = ( φ e ′ ψ e ′ e ′ ) a ( eφ e ψ e )= e ′ ae = a. Moreover, for any a ∈ ǫAǫ we have, using a = ǫaǫ and Proposition 1.4, θ e ′ e (cid:0) η e ′ e ( aM e ′ e ) (cid:1) = θ e ′ e ( φ e ′ aψ e )= ψ e ′ φ e ′ aψ e φ e M e ′ e = ( ψ e ′ φ e ′ ǫ ) a ( ǫψ e φ e ) M e ′ e = ǫaǫ = a. We now want to extend θ e ′ e and η e ′ e to algebra isomorphisms. We have a direct sumdecomposition ˆ ǫA ˆ ǫ = M e ′ ,e ∈I ǫ e ′ Ae. (1.11)We define two maps θ ǫ : ˆ ǫA ˆ ǫ → Mat I ǫ ( ǫAǫ ) ,η ǫ : Mat I ǫ ( ǫAǫ ) → ˆ ǫA ˆ ǫ, by θ ǫ := M e ′ ,e ∈I ǫ θ e ′ e ,η ǫ := M e ′ ,e ∈I ǫ η e ′ e . These two maps are inverse isomorphisms of K -modules. To prove that they are inverse iso-morphism of K -algebras, it suffices to prove that θ ǫ is a morphism of K -algebras. Recallingthe decomposition (1.11), it suffices to prove that θ ǫ ( a a ) = θ ǫ ( a ) θ ǫ ( a ) , (1.12) or any a i ∈ e ′ i Ae i for any e i ∈ I ǫ . If e = e ′ then the left-hand side is zero, and so is theright-hand one since M e ′ e M e ′ e = 0 Mat I ǫ ( ǫAǫ ) . Thus, we now assume that e = e ′ . Wehave a = a e and a a = a e a ∈ e ′ Ae , thus using (1.1b) we obtain θ ǫ ( a a ) = θ e ′ e ( a a )= ψ e ′ a ( e ) a φ e M e ′ e = ψ e ′ a ( e φ e ψ e ) a φ e M e ′ e = ( ψ e ′ a e φ e )( ψ e a φ e ) M e ′ e = (cid:0) ψ e ′ a φ e M e ′ e (cid:1) ( ψ e a φ e M e e )= θ e ′ e ( a ) θ e e ( a )= θ ǫ ( a ) θ ǫ ( a ) . This concludes the proof.
Corollary 1.13.
Assume that for all ǫ, ǫ ′ ∈ J we have ǫ = ǫ ′ = ⇒ ˆ ǫA ˆ ǫ ′ = { } . (1.14) Then have the following isomorphism of K -algebras: A ≃ M ǫ ∈J Mat I ǫ ( ǫAǫ ) . Proof.
The assumption (1.14) implies that A ≃ M ǫ ∈J ˆ ǫA ˆ ǫ. We now use the result of Theorem 1.10.
We here review and generalise the decomposition theorem from [21, §6] to the case of apossibly infinite quiver. A careful analysis of the proofs in this section will be the startingpoint of several proofs later in the paper.
Let Γ be a loop-free quiver, possibly infinite. We write I (respectively A ) for the vertex(resp. arrow) set. We have a map A → I × I given by A ∋ a (cid:0) o ( a ) , t ( a ) (cid:1) ∈ I × I . Theloop-free condition says that for all a ∈ A we have o ( a ) = t ( a ). For any i, j ∈ I , we write | i → j | for the (finite) number of a ∈ A such that o ( a ) = i and t ( a ) = j . We also define i · j := | i → j | + | i ← j | . (We warn the reader that the usual quantity is − i · j .) For any i, j ∈ I we define d ( i, j ) := ( i · j, if i = j, − , otherwise . Let u, v be two indeterminates over K . For any i, j ∈ I , we define the polynomial Q ij ( u, v ) ∈ K [ u, v ] by Q ij ( u, v ) := ( ( − | i → j | ( u − v ) i · j , if i = j, , otherwise, (2.1)Note that Q ij ( u, v ) = Q ji ( v, u ) = Q ij ( − v, − u ) . (2.2)Let n ∈ N and S n be the symmetric group on n letters. We denote by r a the transposition( a, a + 1) ∈ S n for any a ∈ { , . . . , n − } . We will consider the following two actions of S n : the natural action on { , . . . , n } , given by r a · i := r a ( i ) for all a ∈ { , . . . , n − } and i ∈ { , . . . , n } ;• the action on I n by place permutation, given by r a · ( . . . , i a , i a +1 , . . . ) := ( . . . , i a +1 , i a , . . . ) , (2.3)for any i = ( i , . . . , i n ) ∈ I n and a ∈ { , . . . , n − } .Let α ⊆ I n be a finite S n -stable subset, that is, a finite union of S n -orbits. Definition 2.4 (Khovanov–Lauda [16, 17], Rouquier [23]) . The quiver Hecke algebra asso-ciated with the quiver Γ and the finite stable S n -subset α ⊆ I n , denoted by R α (Γ), is theassociative unitary K -algebra generated by elements { y a } ≤ a ≤ n ∪ { ψ b } ≤ b ≤ n − ∪ { e ( i ) } i ∈ α , and relations, for any i , j ∈ α and a, b ∈ { , . . . , n } , X i ∈ α e ( i ) = 1 , e ( i ) e ( j ) = δ ij e ( i ) , y a y b = y b y a , y a e ( i ) = e ( i ) y a , (2.5)and ψ a e ( i ) = e ( r a · i ) ψ a , (2.6)( ψ a y b − y r a ( b ) ψ a ) e ( i ) = − e ( i ) , if b = a and i a = i a +1 ,e ( i ) , if b = a + 1 and i a = i a +1 , , otherwise, (2.7)if a ≤ n −
1, and finally ψ a ψ b = ψ b ψ a , if | a − b | > , (2.8) ψ a e ( i ) = Q i a i a +1 ( y a , y a +1 ) e ( i ) , (2.9)( ψ b +1 ψ b ψ b +1 − ψ b ψ b +1 ψ b ) e ( i ) = Q i b i b +1 ( y b , y b +1 ) − Q i b i b +1 ( y b +2 , y b +1 ) y b − y b +2 e ( i ) , if i b = i b +2 , , otherwise , (2.10)if a ≤ n − b ≤ n − R n (Γ) := L α R α (Γ), where α runs over all the orbits of I n under the action of S n . If Γ is finite, the direct sum is finite and R n (Γ) is a unitary algebra,with unit P i ∈ I n e ( i ). Note that if n = 0 then R α (Γ) = R (Γ) = K . Proposition 2.11 ([16, 17, 23]) . The algebra R α (Γ) is endowed with the Z -grading givenby deg e ( i ) = 0 , deg y a = 2 , deg ψ b e ( i ) = d ( i b , i b +1 ) , for all i ∈ α and a, b ∈ { , . . . , n } with b ≤ n − . For any w ∈ S n , choose a reduced expression w = r a · · · r a k and define ψ w := ψ a · · · ψ a k .Note that the element ψ w may depend on the chosen reduced expression. Proposition 2.12 ([16, 17, 23]) . The algebra R α (Γ) is a free K -module, and { y a · · · y a n n ψ w e ( i ) : a i ∈ N , w ∈ S n , i ∈ α } , is a K -basis.Remark . We recall that there is a one-to-one correspondence between S n -orbits α ⊂ I n and maps ˆ α : I → N of weight n , namely such that P i ∈ I ˆ α ( i ) = n (the number ˆ α ( i ) countsthe number of occurrence of i in any element in the orbit α ). .2 Disjoint union of quivers Let d ∈ N ∗ . Like in [21, §6.1.3], we assume that the quiver Γ decomposes as a disjoint unionof full subquivers Γ = d G j =1 Γ ( j ) , where there are no arrows between Γ ( j ) and Γ ( j ′ ) if j = j ′ . We denote by I = ∐ dj =1 I ( j ) thesubsequent partition of the vertex set. Note that Q ii ′ = 1 whenever i ∈ I ( j ) and i ′ ∈ I ( j ′ ) with j = j ′ .Now we consider a special class of finite unions of S n -orbits in I n . We let G be a finitegroup acting on I and, for each j ∈ { , . . . , d } , we assume that I ( j ) is stable under the actionof G . We denote G n = G n ⋊ S n , the semi-direct product where S n acts on place permutation on G n .The semidirect product G n acts naturally on I n . For any g = ( g , . . . , g n ) ∈ G n and w ∈ S n we have, for all ( i , . . . , i n ) ∈ I n ,( g, w ) · ( i , . . . , i n ) = (cid:0) g · i w − (1) , . . . , g n · i w − ( n ) (cid:1) . We fix α ⊆ I n to be a G n -orbit. Note that α is indeed a finite S n -stable subset of I n asin §2.1. For any i ∈ α and j ∈ { , . . . , d } , let i ( j ) be the tuple obtained from i by removing theentries that are not in I ( j ) . We denote by n j ( i ) the number of remaining entries, that is, thenumber of components of i ( j ) . It follows easily from the fact that each I ( j ) is stable underthe action of G that:the tuple ( n ( i ) , . . . , n d ( i )) is the same for each i ∈ α . (2.14)Thus, we denote, for each j ∈ { , . . . , d } , by n j ( α ) the unique value of n j ( i ) for i ∈ α .We may simply write n j instead of n j ( α ) when α is clear from the context. Note that n + · · · + n d = n .We define α ( j ) := n i ( j ) : i ∈ α o ⊆ ( I ( j ) ) n j . The set α ( j ) is a finite S n j -stable subset of ( I ( j ) ) n j . We will see in (2.17) that it is in facta G n j -orbit.In addition to (2.14), we will need the following property of α . Proposition 2.15.
Recall that α ⊆ I n is a G n -orbit. We have: α (1) × · · · × α ( d ) ⊂ α . (2.16) where we use implicitly the natural identification (by concatenation) of I n × · · · × I n d witha subset of I n .Proof. Let us provide a proof which shows all the various elements explicitly. Since α is a G n -orbit, it can be written of the form: α = { (cid:0) g · i w − (1) , . . . , g n · i w − ( n ) (cid:1) | g , . . . , g n ∈ G , w ∈ S n } , for some element ( i , . . . , i n ) ∈ I n . By invariance under S n , we can choose ( i , . . . , i n ) in anordered form as follows: (cid:0) i , . . . , i n , . . . . . . , i d , . . . , i dn d (cid:1) , here i jk ∈ I ( j ) for all j ∈ { , . . . , d } and k ∈ { , . . . , n j } . Then it is clear that for each j ∈ { , . . . , d } , we have simply α ( j ) = n(cid:0) g · i jw − (1) , . . . , g n j · i jw − ( n j ) (cid:1) (cid:12)(cid:12) g , . . . , g n j ∈ G , w ∈ S n j o . Property (2.16) is now immediate to check.From the proof of the preceding proposition, it is easy to see that the map (cid:8) G n -orbits of I n (cid:9) −→ G n ,...,n d ≥ n + ··· + n d = n d Y j =1 (cid:8) G n j -orbits of (cid:0) I ( j ) (cid:1) n j (cid:9) , (2.17)given by α (cid:0) α (1) , . . . , α ( d ) (cid:1) is a bijection. The inverse map associates to (cid:0) α (1) , . . . , α ( d ) (cid:1) the smallest G n -stable subset in I n containing α (1) × · · · × α ( d ) . Remark . What we actually need for the results of this section is a subset α satisfyingproperties (2.14) and (2.16). However, since we will use in all the paper only G n -orbits, wefind it more convenient to start directly with G n -orbits. In fact we will only use the groups G = { } and G = Z / Z , but considering an arbitrary finite group G does not lead to anycomplication. Remark . • Let Ω be the set of G -orbits of I . Generalising Remark 2.13, it is easy to seethat there is a one-to-one correspondence between G n -orbits α ⊆ I n and maps ˆ α : Ω → N such that P ω ∈ Ω ˆ α ( ω ) = n . If α ⊆ I n is a G n -orbit and ω ∈ Ω, then ˆ α ( ω ) counts the numberof occurrence of the elements of ω in any element of α . • For each j = 1 , . . . , d , let Ω ( j ) be the set of G -orbits of I ( j ) . We have Ω = ∐ dj =1 Ω ( j ) .Then the bijection (2.17) in terms of maps simply associates to ˆ α : Ω → N the restrictionsˆ α | Ω ( j ) : Ω ( j ) → N to each Ω ( j ) . Example . Let us give an example of a subset α not satisfying property (2.16). Let n = 2 and α = { ( a, A ) , ( A, a ) , ( b, B ) , ( B, b ) } where a, b ∈ I (1) and A, B ∈ I (2) . Then α isa union of two S -orbits and it satisfies (2.14). It does not satisfy (2.16). Indeed, we have α (1) = { a, b } and α (2) = { A, B } but, for example, ( a, B ) / ∈ α . We keep α ⊆ I n a G n -orbit for some finite group G acting on each set I ( j ) . We may (andwe will) simply write n j instead of n j ( α ).For each i ∈ I , we set p ( i ) = j ∈ { , . . . , d } if i ∈ I ( j ) . Then for each i = ( i , . . . , i n ) ∈ I n ,we define its profile by p ( i ) = (cid:0) p ( i ) , . . . , p ( i n ) (cid:1) ∈ { , . . . , d } n . LetProf α := { p ( i ) , i ∈ α } ⊆ { , . . . , d } n be the set of all profiles of elements of α . Note that (2.14) ensures that Prof α is also a singleorbit, now for the action of S n on { , . . . , d } n by place permutation.A natural element to consider in this orbit Prof α is t α := (1 , . . . , , , . . . , , . . . . . . , d, . . . , d ) , where each j ∈ { , . . . , d } appears exactly n j times. Then every element t ∈ Prof α can bereordered to obtain the distinguished element t α . More precisely, for any t ∈ Prof α , theset of elements w ∈ S n such that w · t = t α forms a right coset in S n for the subgroup S n × · · · × S n d (the stabiliser of t α ). There is a unique minimal length element in thiscoset (see e.g. [10]) and we denote it π t . In particular, the element π t is the unique minimallength element of S n such that π t · t = t α .For any t ∈ Prof α , we define the idempotent e ( t ) = X i ∈ αp ( i )= t e ( i ) ∈ R α (Γ) , nd we set I := (cid:8) e ( t ) : t ∈ Prof α (cid:9) . It is a complete set of orthogonal idempotent and its cardinality is (cid:0) nn ,...,n d (cid:1) . Then, for any t ∈ Prof α we fix a reduced expression π t = r a · · · r a k and define ψ t := ψ a · · · ψ a k ∈ R α (Γ) , (2.21a) φ t := ψ a k · · · ψ a ∈ R α (Γ) . (2.21b)In the following proposition, the grading on Mat I (cid:0) e ( t α ) R α (Γ) e ( t α ) (cid:1) is trivially inducedfrom the grading on e ( t α ) R α (Γ) e ( t α ) (an homogeneous element of degree N is a matrixwhere all coefficients are homogeneous elements of degree N ). Proposition 2.22.
We have an isomorphism of graded algebras: R α (Γ) ≃ Mat( nn ,...,nd ) (cid:0) e ( t α ) R α (Γ) e ( t α ) (cid:1) . Proof.
The proof follows the same steps as in [21] and we only give a sketch and the precisereferences to [21]. First we have that the data { e ( t ) , ψ t , φ t } t ∈ Prof α in R α (Γ) enters thegeneral setting (1.1) of Section 1, namely we have, for any t ∈ Prof α (see [21, Proposition6.18]), φ t ψ t e ( t ) = e ( t ) φ t ψ t = e ( t ) . (2.23)The main point to prove (2.23) is the following fact: ψ a e ( t ) = e ( t ) , (2.24)for any a ∈ { , . . . , n − } and t ∈ Prof α such that t a = t a +1 (see [21, Lemma 6.15]). Similarly,we obtain, for any t ∈ Prof α , ψ t e ( t ) φ t = ψ t φ t e ( t α ) = e ( t α ) . (2.25)This last equality ensures that the set J in the notation of §1 is J = { e ( t α ) } . Since J isreduced to one element, we deduce that the assumption (1.14) is automatically satisfied, andwe can use Corollary 1.13 to obtain the proposition. Finally, the fact that the isomorphismis homogeneous follows from deg ψ t e ( t ) = deg φ t e ( t ) = 0 for any t ∈ Prof α (see [21, Remark6.29]). Remark . Similarly to (2.24), we have (see [21, Lemma 6.20]) y a φ t e ( t ) = φ t y π t ( a ) e ( t ) , (2.27)for any a ∈ { , . . . , n − } and t ∈ Prof α such that t a = t a +1 , and also (see [21, Lemma6.15]) ψ a +1 ψ a ψ a +1 e ( t ) = ψ a ψ a +1 ψ a e ( t ) , (2.28)for any a ∈ { , . . . , n − } and t ∈ Prof α such that t a = t a +2 . In particular, (2.28) impliesthat the quantities ψ t e ( t ) and e ( t ) φ t do not depend on the chosen reduced expression for π t . We now want to write the algebra e ( t α ) R α (Γ) e ( t α ) as a tensor product. Recall that α isa G n -orbit and thus satisfies properties (2.14) and (2.16). We have already used the firstproperty. The second will be explicitly used during the proof of the next result.Note that, for any j ∈ { , . . . , d } , the algebra R α ( j ) (Γ ( j ) ) is well-defined since α ( j ) consistsof n j -tuples of vertices I ( j ) of Γ ( j ) and is stable under permutations (see §2.2.1). Theorem 2.29.
We have an (explicit) isomorphism of graded algebras: e ( t α ) R α (Γ) e ( t α ) ≃ R α (1) (Γ (1) ) ⊗ · · · ⊗ R α ( d ) (Γ ( d ) ) . roof. We construct an algebra homomorphism f from the tensor product to e ( t α ) R α (Γ) e ( t α )as follows. For any i ( j ) ∈ α ( j ) ⊆ ( I ( j ) ) n j with j ∈ { , . . . , d } we define f (cid:0) e ( i (1) ) ⊗ · · · ⊗ e ( i ( d ) ) (cid:1) := e ( i (1) , . . . , i ( d ) ) . Note that (cid:0) i (1) , . . . , i ( d ) (cid:1) ∈ α due to Proposition 2.15. Moreover, for any j ∈ { , . . . , d } wedenote y ( j ) a and ψ ( j ) b the generators of R α ( j ) (Γ ( j ) ) in the tensor product and we define f ( y ( j ) a ) := e ( t α ) y n + ··· + n j − + a e ( t α ) ,f ( ψ ( j ) b ) := e ( t α ) ψ n + ··· + n j − + b e ( t α ) , for all a, b ∈ { , . . . , n j } with b ≤ n j −
1. By [21, Lemma 6.24], the map f is indeed ahomomorphism. Using the basis of Proposition 2.12, we can prove that f sends a basis ontoa basis and thus is an isomorphism (see [21, Proposition 6.25]). Finally, the isomorphism f is clearly homogeneous.Combining Theorem 2.29 with Proposition 2.22, we obtain the main result of this section. Corollary 2.30.
We have an (explicit) isomorphism of graded algebras: R α (Γ) ≃ Mat( nn ,...,nd ) d O j =1 R α ( j ) (Γ ( j ) ) . Remark . If α = ∐ ki =1 α i the decomposition of α into S n -orbits, then we have R α (Γ) = ⊕ ki =1 R α i (Γ). So of course, as far as the algebras R α (Γ) are concerned, taking α a single S n -orbit would be enough. However, we really needed a more general setting since we willapply later the results above for orbits α ⊂ I n of the Weyl group of type B.We now show how to recover [21, Theorem 6.26], with the difference that the result weobtain here is also valid if the quiver Γ is infinite. Corollary 2.32.
We have an (explicit) isomorphism of graded algebras: R n (Γ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) d O j =1 R n j (Γ ( j ) ) . Proof.
We write I n / S n to denote the S n -orbits in I n . We apply the isomorphism of Corol-lary 2.30 in each term of right-hand side of the equality R n (Γ) = ⊕ α R α (Γ), where α runs over I n / S n (so we use the situation G = { } here). Recalling the 1:1-correspondence in (2.17),we obtain R n (Γ) ≃ M α ∈ I n / S n R α (Γ) ≃ M α ∈ I n / S n Mat( nn α ) ,...,nd ( α ) ) d O j =1 R α ( j ) (Γ ( j ) ) ≃ M n ,...,n d ≥ n + ··· + n d = n M α ∈ I n / S n n j ( α )= n j Mat( nn α ) ,...,nd ( α ) ) d O j =1 R α ( j ) (Γ ( j ) ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) M α ∈ I n / S n n j ( α )= n j d O j =1 R α ( j ) (Γ ( j ) ) M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) M α (1) ∈ I n / S n · · · M α ( d ) ∈ I nd / S nd d O j =1 R α ( j ) (Γ ( j ) ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) d O j =1 R n j (Γ ( j ) ) , as desired. We keep the above setting with the quiver Γ, its full subquivers Γ ( j ) and a G n -orbit α . Inaddition, let Λ = (Λ i ) i ∈ I be a finitely-supported family of non-negative integers. Definition 2.33 ([23, 6]) . The cyclotomic quiver Hecke algebra R Λ α (Γ) is the quotient ofthe quiver Hecke algebra R α (Γ) by the two-sided ideal I Λ α generated by the relations y Λ i e ( i ) = 0 , (2.34)for all i = ( i , . . . , i n ) ∈ α .Since the above relations are homogeneous, the cyclotomic quiver Hecke algebras isgraded, as in Proposition 2.11. Note that if Λ i = 0 for all i then R Λ α (Γ) = ( { } , if n ≥ ,K, if n = 0 . As in [21, §6.4.1], we want to state Corollaries 2.30 and 2.32 in the cyclotomic setting.First, for any j ∈ { , . . . , d } let Λ ( j ) be the restriction of Λ to I ( j ) . Theorem 2.35.
We have an (explicit) isomorphism of graded algebras: R Λ α (Γ) ≃ Mat( nn ,...,nd ) d O j =1 R Λ ( j ) α ( j ) (Γ ( j ) ) . Proof.
The proof is similar to the one of [21, Theorem 6.30]. We provide details since it willbe used later in the paper.Note that ⊗ dj =1 R Λ ( j ) α ( j ) (Γ ( j ) ) is the quotient of ⊗ dj =1 R α ( j ) (Γ ( j ) ) by the two-sided ideal I Λ α, ⊗ := h ⊗ · · · ⊗ ⊗ I Λ ( j ) α ( j ) ⊗ ⊗ · · · ⊗ , j = 1 , . . . , d i generated by the ideals I Λ ( j ) α ( j ) in position j in the tensor product. We will identify the algebra ⊗ dj =1 R α ( j ) (Γ ( j ) ) with the algebra e ( t α ) R α (Γ) e ( t α ) thanks to the explicit isomorphism givenin the proof of Theorem 2.29. With this identification, the ideal I Λ α, ⊗ is generated by theelements y Λ ib b e ( i ) , where i ∈ α is of profile t α , and b is of the form b = n + · · · + n j − + 1 for j ∈ { , . . . , d } .Now let θ be the isomorphism of Proposition 2.22 and η its inverse. For convenience wedenote during the proof N := (cid:0) nn ,...,n d (cid:1) . We will prove the following two inclusions: θ (cid:0) I Λ α (cid:1) ⊆ Mat N (cid:0) I Λ α, ⊗ (cid:1) , (2.36a) I Λ α ⊇ η (cid:0) Mat N (cid:0) I Λ α, ⊗ (cid:1)(cid:1) . (2.36b)Let t ′ , t ∈ Prof α . First recall that for h ∈ e ( t ′ ) R α (Γ) e ( t ), we have θ ( h ) = ψ t ′ hφ t M t ′ t ∈ Mat N ( e ( t α ) R α (Γ) e ( t α )) . hile for h ∈ e ( t α ) R α (Γ) e ( t α ) we have η ( hM t ′ t ) = φ t ′ hψ t , where the elements φ t , ψ t were introduced in (2.21). • Let i ∈ α of profile t . By (2.23), (2.6) and (2.27) we have: y Λ i e ( i ) = y Λ i e ( i ) e ( t ) = y Λ i e ( i ) φ t ψ t e ( t ) = φ t y Λ i π t (1) e ( π t · i ) ψ t e ( t ) . Thus, to prove (2.36a) it suffices to show that θ (cid:16) y Λ i π t (1) e ( π t · i ) (cid:17) ∈ Mat N (cid:0) I Λ α, ⊗ (cid:1) . By definition of π t , we have that i ′ := π t · i has profile t α and therefore y Λ i π t (1) e ( i ′ ) ∈ e ( t α ) R α (Γ) e ( t α ). Let b := π t (1) so that we have i = i ′ b , and moreover, by [21, Proposition6.7], the element b is of the form n + · · · + n j − + 1. We conclude that θ (cid:16) y Λ i π t (1) e ( π t · i ) (cid:17) = y Λ i ′ b b e ( i ′ ) M t α t α ∈ Mat N (cid:0) I Λ α, ⊗ (cid:1) . • Let i ∈ α with profile t α and let b = n + · · · + n j − + 1 with j ∈ { , . . . , d } such that n j = 0. Let us prove that η (cid:16) y Λ ib b e ( i ) M t ′ t (cid:17) ∈ I Λ α . Since M t ′ t = M t ′ t ′′ M t ′′ t for any t ′′ it is enough to prove it for a single value of t ′ . So withoutloss of generality, since n j = 0 we can assume that t ′ starts with j so that π t ′ (1) = b . Weconclude that η (cid:16) y Λ ib b e ( i ) M t ′ t (cid:17) = φ t ′ y Λ ib b e ( i ) ψ t = y Λ ib e ( π − t ′ · i ) φ t ′ ψ t ∈ I Λ α , since, if we denote i ′ = π − t ′ · i then we have i ′ = i b .This concludes the proof of (2.36) showing that we have θ (cid:0) I Λ α (cid:1) = Mat N (cid:0) I Λ α, ⊗ (cid:1) . Thus we can deduce the isomorphism of Theorem 2.35 from Corollary 2.30.
Remark . • We saw that if Λ ( j ) ≡ ( j ) for some j then, if moreover α ( j ) = ∅ (that is, if n j ( α ) = 0), we have R Λ ( j ) α ( j ) (Γ ( j ) ) = { } from the defining relations. So inturn, Theorem 2.35 implies that R Λ α (Γ) = { } .• The conclusion of the preceding item can in fact be seen more directly. Indeed thecyclotomic relations in R Λ α (Γ) imply that e ( i ) = 0 for all i ∈ α with i ∈ Γ ( j ) . So wehave that the idempotent e ( t ) is 0 for any profile t starting with j (and at least oneprofile like this exists in Prof α when n j ( α ) = 0). Since: ψ t e ( t ) φ t = e ( t α ) and φ t e ( t α ) ψ t = e ( t ) , it follows immediately that if n j ( α ) = 0 then all idempotents e ( t ) are 0 and in turn allidempotents e ( i ), i ∈ α , are 0, which shows that R Λ α (Γ) = { } .As in Corollary 2.32, we deduce the following corollary. Corollary 2.38.
We have an (explicit) isomorphism of graded algebras: R Λ n (Γ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) d O j =1 R Λ ( j ) n j (Γ ( j ) ) . emark . It follows from Remark 2.37 that we can assume that Λ is supported on allcomponents of Γ, that is Λ ( j ) j ∈ { , . . . , d } . In other words, we can replacefrom the beginning Γ by ˜Γ where we removed the components Γ ( j ) such that Λ ( j ) ≡
0. Inparticular, we have R Λ n (Γ) = R Λ | ˜ I n (˜Γ), where ˜ I denotes the vertex set of ˜Γ. We could havedone that but it turned out to be not really necessary to state Theorem 2.35 or Corollary 2.38.For example, in Corollary 2.38, if Λ ( j ) ≡ j then all the summands with n j = 0are { } and can thus be removed from the direct sum. The aim of this section is to unite the definitions of quiver Hecke algebras for type B thatare introduced in [25] by Varagnolo and Vasserot and in [19, 20] by the first author andWalker.
Let Γ be a quiver as in §2.1. We also adopt the notation of this subection. Let θ be aninvolution of Γ, that is, the map θ is an involution on both sets I and A and satisfies θ ( o ( a )) = t ( θ ( a )) , (3.1)for all a ∈ A . Note the following consequence: for any i, j ∈ I we have | i → j | = | θ ( j ) → θ ( i ) | and thus i · j = θ ( i ) · θ ( j ) . (3.2)It follows from the definition (2.1) of the polynomials Q ij and from (3.1) again that Q ij ( u, v ) = Q θ ( j ) θ ( i ) ( u, v ) , (3.3)for any i, j ∈ I .Let B n be the group of signed permutations of {± , . . . , ± n } , that is, the group ofpermutations π of {± , . . . , ± n } satisfying π ( − i ) = − π ( i ) for all i ∈ { , . . . , n } . We havea natural isomorphism B n ≃ ( Z / Z ) n ⋊ S n . In particular we are in the setting of §2.2with G = Z / Z , which acts on I via the canonical surjection G ։ h θ i . We have a naturalinclusion S n ⊆ B n , where r a is identified with ( a, a +1)( − a, − a −
1) for all a ∈ { , . . . , n − } .We see B n as a Weyl group of type B by adding the generator r := ( − , B n on I n is given by (2.3) and r · ( i , . . . , i n ) := ( θ ( i ) , i , . . . , i n ) , for any i = ( i , . . . , i n ) ∈ I n . Let β ⊆ I n be a B n -orbit. In particular, the set β is a finite S n -stable subset of I n . Remark . The result of Remark 2.19 can here be written as follows. There is a one-to-onecorrespondence between B n -orbits β ⊂ I n and maps ˆ β : I → N such that ˆ β = ˆ β ◦ θ and P i ∈ Iθ ( i ) = i ˆ β ( i ) + P i ∈ Iθ ( i )= i ˆ β ( i ) = n (the number ˆ β ( i ) counts the number of occurrence of both i and θ ( i ) in any element in the orbit β ). See also [19, Remark 2.5].Let λ ∈ N I and γ ∈ K I . Define d ( i ) := ( λ i + λ θ ( i ) , if γ i = 0 , − , otherwise . For any i ∈ I , we make the following assumptions: θ ( i ) = i = ⇒ γ i = 0 , (3.5a) γ i = 0 = ⇒ [ θ ( i ) = i or d ( i ) = 0] . (3.5b)Note that γ is θ -invariant, that is, we have γ θ ( i ) = γ i , for all i ∈ I. (3.6) emark . • Condition (3.5b) may seem strong; without it we encounter in §A.1 uselesscomplications for our means (see also Remark A.5). • Similarly, one could consider a more general definition than the one below. As forexample in [23, §3.2], we could remove any reference to a quiver and start only with afamily of polynomials associated to the set I with involution θ (namely, Q ij [ u, v ] and apolynomial replacing ( − λ θ ( i y d ( i )1 in the definition below). Then one should look forconditions ensuring the existence of a polynomial representation. We do not pursue in thisdirection to avoid adding another layer of technicalities. Definition 3.8.
The algebra V β (Γ , λ, γ ) is the unitary associative K -algebra generated byelements { y a } ≤ a ≤ n ∪ { ψ b } ≤ b ≤ n − ∪ { e ( i ) } i ∈ β , with the relations (2.5)–(2.10) of Section 2 involving all the generators but ψ , together with ψ e ( i ) = e ( r · i ) ψ , (3.9) ψ ψ b = ψ b ψ , for all b ∈ { , . . . , n − } , (3.10)( ψ y + y ψ ) e ( i ) = 2 γ i e ( i ) , (3.11) ψ y a = y a ψ , for all a ∈ { , . . . , n } , (3.12) ψ e ( i ) = ( ( − λ θ ( i y d ( i )1 e ( i ) , if γ i = 0 , , otherwise , (3.13) (cid:0) ( ψ ψ ) − ( ψ ψ ) (cid:1) e ( i ) = (3.14) ( − λ θ ( i ( − y ) d ( i − y d ( i y + y ψ e ( i ) , if γ i = 0 and θ ( i ) = i ,γ i Q i i ( y , − y ) − Q i i ( y ,y ) y y ( y ψ − γ i ) e ( i ) otherwise,for all i ∈ β .It is clear that the fraction in the first line of the right hand side in (3.14) is a polynomialin y , y . Then we note that the second line in the right hand side of (3.14) is 0 when γ i = 0or when i = i (recalling (2.1)), and is a polynomial in y , y when γ i = 0. So for thesecond line, if γ i = 0 = γ i and i = i then by (3.5a) we have θ ( i ) = i and θ ( i ) = i ,and thus we can use (2.2) and (3.3) so that Q i i ( u, − v ) − Q i i ( u, v ) uv = Q i i ( v, − u ) − Q θ ( i ) θ ( i ) ( u, v ) uv = Q i i ( v, − u ) − Q i i ( u, v ) uv = Q i i ( v, − u ) − Q i i ( v, u ) uv , is a polynomial.Finally, note that when n = 0 then V β (Γ , λ, γ ) = K . Remark . Since β is a finite S n -stable subset of I n , we can also consider the algebra R β (Γ) as defined in §2.1. The subalgebra of V β (Γ , λ, γ ) generated by all the generators but ψ is an obvious quotient of R β (Γ) (see also Corollary 3.28). Remark . If θ has no fixed point in I then V β (Γ , λ, γ ) is exactly the algebra definedin [25]. In this case, by (3.5a) we necessarily have γ i = 0 for any i and (3.5b) is automaticallysatisfied. In particular, in (3.14) the second line is always zero in this situation. Remark . Assume that K is field of characteristic different from 2 and let p, q ∈ K × with q = 1 = p . Let θ : K × → K × be the scalar inversion. For any x ∈ K × , we definethe set I x := { x ǫ q l : ǫ ∈ {± } , l ∈ Z } . Let x , . . . , x k ∈ K × such that the sets I x a arepairwise disjoint. Let Γ be the quiver with vertices I := ∐ ka =1 I x a and arrows between v and q v for all v ∈ I . Finally let λ be the indicator function of P := {± p } ∩ I and define γ i := 1if θ ( i ) = i and γ i := 0 otherwise (thus (3.5) is satisfied). Then V β (Γ , λ, γ ) is exactly the lgebra V I β x defined in [19]. This is, together with the next remark, the situation relevantfor the applications to affine Hecke algebras, see Section 6. Remark . The algebra of [20, §3.1] is obtained with the same choice of Γ , θ as in thepreceding remark, together with γ i := 0 and λ i := 0 for all i . In particular, Condition (3.5b)is satisfied since d ( i ) = 0 for all i ∈ I . We will come back to this particular situation inSection 5.The algebra V β (Γ , λ, γ ) is endowed with the Z -grading given bydeg e ( i ) = 0 , (3.19a)deg y a = 2 , (3.19b)deg ψ e ( i ) = d ( i ) , (3.19c)deg ψ b e ( i ) = d ( i b , i i +1 ) . (3.19d)The homogeneity of the defining relations that do not involve ψ is as in Section 2, theother ones being a simple calculation. For (3.11) note that if γ i = 0 there is nothing to check,and if γ i = 0 then by definition we have d ( i ) = − ψ y e ( i ) = deg y ψ e ( i ) = 0.To check the last relation, let us write i i instead of i and even a instead of i a and ¯ a insteadof θ ( i a ) . We have (cid:0) ( ψ ψ ) − ( ψ ψ ) (cid:1) e (12) = ψ ψ ψ ψ e (12) − ψ ψ ψ ψ e (12)= ψ e (1¯2) ψ e (¯21) ψ e (21) ψ e (12) − ψ e (¯2¯1) ψ e (2¯1) ψ e (¯12) ψ e (12) . (3.20)We have: deg ψ e (1¯2) = deg ψ e (12) = d (1) , deg ψ e (21) = deg ψ e (2¯1) = d (2) . Moreover, by (3.2) we have deg ψ e (¯21) = d (¯2 ,
1) = d (1 , ¯2) = d (¯1 ,
2) = deg ψ e (¯12) , deg ψ e (12) = d (1 ,
2) = d (2 ,
1) = d (¯2 , ¯1) = deg ψ e (¯2¯1) . Thus, the quantity (cid:0) ( ψ ψ ) − ( ψ ψ ) (cid:1) e ( i ) is homogeneous of degree d ( i ) + d ( i ) + d (cid:0) i , i (cid:1) + d (cid:0) i , θ ( i ) (cid:1) . A quick calculation now shows that the last relation is homogeneous (note that in the firstcase we have γ i = 0 by (3.6)). We now want to give an analogue of the basis theorem Proposition 2.12 for quiver Heckealgebras. As in [16, 17, 23], we will construct a polynomial realisation of V β (Γ , λ, γ ) . Let ( P ij ( u, v )) i,j ∈ I be a family of polynomials satisfying P ij ( u, v ) = P ij ( − v, − u ) , (3.21a) P ij ( u, v ) = P θ ( j ) θ ( i ) ( u, v ) , (3.21b)and such that P ij ( u, v ) P ji ( v, u ) = Q ij ( u, v ) . (3.22)Note that P ij ( u, v ) := ( u − v ) | j → i | if i = j and P ij ( u, v ) := 0 if i = j is an example of sucha family, by (3.1). Now let ( α i ( y )) i ∈ I be a family of polynomials such that α θ ( i ) ( y ) α i ( − y ) = ( − λ θ ( i ) y d ( i ) , if γ i = 0 , (3.23) α i ( y ) = 0 , otherwise . (3.24) ote that if γ i = 0 we can just set α i ( y ) := y λ θ ( i ) . We now consider the sum of polyno-mials algebras K [ x, β ] := ⊕ i ∈ β K [ x , . . . , x n ] i , where i denotes the unit of the summandcorresponding to i , so that f i = i f, for all f ∈ K [ x , . . . , x n ] and i ∈ β, i j = δ ij i , for all i , j ∈ β. The Weyl group B n acts on K [ x , . . . , x n ] by w f ( x , . . . , x n ) := f (cid:0) w − · ( x , . . . , x n ) (cid:1) for any w ∈ B n and f ∈ K [ x , . . . , x n ] , where the action of the generator r on ( x , . . . , x n ) is bymultiplying x by − , and the action of the generator r a , a = 1 , . . . , n − , on ( x , . . . , x n ) is by exchanging x a and x a +1 . The action of B n on K [ x , . . . , x n ] extends by linearity to K [ x, β ] by setting w ⋆ f i := w f w · i for any i ∈ β .We now consider the linear action of V β (Γ , λ, γ ) on K [ x, β ] given on the generators by e ( j ) · f i := δ ij f i = δ ij i f,y a · f i := x a f i = x a i f,ψ b · f i := δ i b ,i b +1 r b f − fx b − x b +1 i + P i b ,i b +1 ( x b +1 , x b ) r b f r b · i = (cid:16) δ i b ,i b +1 ( x b − x b +1 ) − ( r b −
1) + P i b ,i b +1 ( x b +1 , x b ) r b (cid:17) ⋆ f i ,ψ · f i := (cid:18) γ i f − r fx + α i ( x ) r f (cid:19) r · i = (cid:0) γ i x − (1 − r ) + α i ( x ) r (cid:1) ⋆ f i , for any i , j ∈ β and f ∈ K [ x , . . . , x n ] . Lemma 3.25.
The previous action is well-defined.
The proof of Lemma 3.25 is given in Appendix A. For each w ∈ B n we now fix a reducedexpression w = r a · · · r a k and define ψ w := ψ a · · · ψ a k ∈ V β (Γ , λ, γ ) . Note that the element ψ w may depend on the chosen reduced expression. Theorem 3.26.
The algebra V β (Γ , λ, γ ) is a free K -module, and { y a · · · y a n n ψ w e ( i ) : a i ∈ N , w ∈ B n , i ∈ β } , is a K -basis.Proof. As in [16, 17, 23], successively applying the defining relations of V β (Γ , λ, γ ) we cansee that the above family is a spanning set, hence it remains to prove that it is linearlyindependent. For any b ∈ { , . . . , n − } , i ∈ β and f ∈ K [ x , . . . , x n ] we can write ψ b · f i = (cid:16) A r b i r b + A ,r b i (cid:17) ⋆ f i , where A r b i , A ,r b i ∈ K ( x , . . . , x n ) with A r b i non-zero (recall that P ij = 0 if i = j ). If < is theBruhat order on B n , we deduce that for each w ∈ B n we can write ψ w · f i = A w i w + X w ′ End K ( K [ x, β ]) and thus conclude the proof.As a corollary, we obtain the sequel of Remark 3.15. Corollary 3.28. The subalgebra of V β (Γ , λ, γ ) generated by all generators but ψ is isomor-phic to R β (Γ) . Let Γ be a quiver with an involution θ and λ ∈ N I , γ ∈ K I as in §3.1. Let d be a positiveinteger and write Γ = ∐ dj =1 Γ ( j ) such that• each Γ ( j ) is a full subquiver of Γ ;• each Γ ( j ) is stable under θ .We write I = ∐ dj =1 I ( j ) the corresponding partition of the vertex set of Γ . Recall that B n ≃ G n ⋊ S n with G = Z / Z acting on I via G ։ h θ i . In particular, each I ( j ) for j ∈ { , . . . , d } is stable under the action of G so that we are in the setting of §2.2.Let β be a B n -orbit in I n . As explained in §2.2, both properties (2.14) and (2.16) aresatisfied. In particular, for any j ∈ { , . . . , d } we have an integer n j ( β ) = n j and we have a B n j -orbit β ( j ) ⊆ ( I ( j ) ) n j .For any j ∈ { , . . . , d } , we define λ ( j ) ∈ N I ( j ) (respectively γ ( j ) ∈ K I ( j ) ) to be therestriction of λ (resp. γ ) to I ( j ) . Theorem 4.1. We have an (explicit) isomorphism of graded algebras V β (Γ , λ, γ ) ≃ Mat( nn ,...,nd ) d O j =1 V β ( j ) (cid:16) Γ ( j ) , λ ( j ) , γ ( j ) (cid:17) . As in §2, will first apply the result of §1 and then prove an isomorphism with a tensorproduct. Parts 4.1 and 4.2 are devoted to the proof of Theorem 4.1, which is a directconsequence of (4.2) and Proposition 4.3. As defined in §2.2.2, to each i ∈ β we associate its profile p ( i ) ∈ { , . . . , d } n , and we write Prof β ⊆ { , . . . , d } n to denote the set of all profiles of elements of β . Any element of Prof β can be reordered so that we obtain t β = (1 , . . . , , . . . , d, . . . , d ) , where each j ∈ { , . . . , d } appears exactly n j times. To any t ∈ Prof β , we define theidempotent e ( t ) := X i ∈ βp ( i )= t e ( i ) ∈ V β (Γ , λ, γ ) , and we define I := { e ( t ) : t ∈ Prof β } . It is a complete set of orthogonal idempotents and its cardinality is exactly (cid:0) nn ,...,n d (cid:1) . Sinceany reduced expression in S n in the generators r , . . . , r n − is also reduced in B n for thesesame generators, the definitions (2.21) make sense in V β (Γ , λ, γ ) for any t ∈ Prof β . More-over, since the defining relations of R β (Γ) are also satisfied in V β (Γ , λ, γ ) , we deduce thatequations (2.23) and (2.25) are still satisfied in V β (Γ , λ, γ ) thus as in §2.2.2 we conclude that V β (Γ , λ, γ ) ≃ Mat( nn ,...,nd ) (cid:0) e ( t β ) V β (Γ , λ, γ ) e ( t β ) (cid:1) . (4.2) .2 Embedding the tensor product The aim of this section is to prove the following proposition. Proposition 4.3. We have an (explicit) isomorphism of graded algebras e ( t β ) V β (Γ , λ, γ ) e ( t β ) ≃ d O j =1 V β ( j ) (cid:16) Γ ( j ) , λ ( j ) , γ ( j ) (cid:17) . Set n = n + · · · + n d . We start by defining a map from the set of generators of the algebra N dj =1 V β ( j ) (cid:0) Γ ( j ) , λ ( j ) , γ ( j ) (cid:1) to e ( t β ) V β (Γ , λ, γ ) e ( t β ) .Let j ∈ { , . . . , d } . We denote ψ ( j )0 , . . . , ψ ( j ) n j − , y ( j )1 , . . . , y ( j ) n j , e ( i j ) with i j ∈ β ( j ) , thegenerators of V β ( j ) (cid:0) Γ ( j ) , λ ( j ) , γ ( j ) (cid:1) . Then we consider the map e ( i ) ⊗ · · · ⊗ e ( i d ) e ( i , . . . , i d ) , (4.4) ψ ( j )0 e ( t β ) ψ n + ··· + n j − . . . ψ ψ ψ . . . ψ n + ··· + n j − e ( t β ) , (4.5) ψ ( j ) a e ( t β ) ψ n + ··· + n j − + a e ( t β ) , a = 1 , . . . , n j − , (4.6) y ( j ) b e ( t β ) y n + ··· + n j − + b e ( t β ) , b = 1 , . . . , n j , (4.7)where each i j ∈ β ( j ) and ( i , . . . , i d ) is simply the concatenation. Note that ( i , . . . , i d ) ∈ β since β is a B n -orbit, using Proposition 2.15. Moreover, the profile of ( i , . . . , i d ) is t β andthus e ( i , . . . , i d ) e ( t β ) = e ( t β ) e ( i , . . . , i d ) = e ( i , . . . , i d ) . By convention, n + · · · + n j − = 0 if j = 1 (and ψ (1)0 ψ ). Note also that the Formula (4.4) extended by linearity gives theimage of an idempotent e ( i j ) ∈ V β ( j ) (cid:0) Γ ( j ) , λ ( j ) , γ ( j ) (cid:1) : e ( i j ) d X j ′ =1 j ′ = j X i j ′ ∈ β ( j ′ ) e ( i , . . . , i d ) . (4.8)Equivalently, the image of e ( i j ) is the sum of the idempotents e ( i ) where the sum is takenover i ∈ β such that the profile of i is t β and moreover ( i n + ··· + n j − +1 , . . . , i n + ··· + n j ) = i j .We will prove that the map given in (4.4)–(4.7) extends to an homomorphism of gradedalgebras denoted ρ and that ρ is bijective. We check that the map given in (4.4)–(4.7) preserves the grading given in (3.19). For theimages of the idempotents and of the generators y ( j ) b , there is nothing to check.Let i j ∈ β ( j ) and i ∈ β such that ( i n + ··· + n j − +1 , . . . , i n + ··· + n j ) = i j . Let a ∈{ , . . . , n j − } . On the one hand, we have deg ψ ( j ) a e ( i j ) = d ( i ja , i ja +1 ) . On the other hand,we have deg ψ n + ··· + n j − + a e ( i ) = d ( i n + ··· + n j − + a , i n + ··· + n j − + a +1 ) = d ( i ja , i ja +1 ) . Finally, on the one hand, we have deg ψ ( j )0 e ( i j ) = d ( i j ) . On the other hand, we claimthat we have deg ψ k . . . ψ . . . ψ k e ( j ) = d ( j k +1 ) , for any k ≥ and any j ∈ β such that j k +1 is not in the same component as j , . . . , j k forthe decomposition of the quiver Γ = ∐ dj =1 Γ ( j ) . Taking k = n + · · · + n j − and j = i , thisconcludes the verification.To prove the claim, we use induction on k . For k = 0 , this is the definition of thedegree of ψ e ( j ) . For k > , we have deg ψ k e ( j ) = j k · j k +1 = | j k → j k +1 | + | j k ← k +1 | = 0 by assumption on j . Similarly, deg ψ k e ( j ′ ) = 0 where j ′ = r k − . . . r . . . r k − r k ( j ) ,since ( j ′ k , j ′ k +1 ) = ( θ ( j k +1 ) , j k ) . It remains to use the induction hypothesis, namely thatdeg ψ k − . . . ψ . . . ψ k − e ( r k ( j )) = d ( j k +1 ) , valid since r k ( j ) has j k +1 in position k . We assume for a moment that the map given in (4.4)–(4.7) extends to an algebra homomor-phism. We denote this map by ρ and we prove here that ρ is bijective.For any j ∈ { , . . . , d } , we write B ( j ) := B n j and we rename its generators to r ( j )0 , . . . , r ( j ) n j − .We recall the following fact. Lemma 4.9. We have an injective group homomorphism B (1) × · · · × B ( d ) → B n ( w , . . . , w d ) w . . . w d given on the generators by, for j ∈ { , . . . , d } , r ( j )0 r n + ··· + n j − . . . r r r . . . r n + ··· + n j − ,r ( j ) a r n + ··· + n j − + a , a = 1 , . . . , n j − . By convention, n + · · · + n j − = 0 if j = 1 (and r (1)0 r ). Moreover, any d -tuple ofreduced expressions is sent onto a reduced expression in B n .Proof. Recall that B n = h r , . . . , r n − i is the group of signed permutations of {± , . . . , ± n } ,with r = (1 , − and r a = ( a, a + 1)( − a, − a − for a = 1 , . . . , n − . Let t := r and t a +1 := r a t a r a for a = 1 , . . . , n − . The element t a corresponds to the transposition ( − a, a ) .For any i ∈ { , . . . , n } and a ∈ { , . . . , i } , we set by convention r a . . . r i − = 1 if a = i .It is easy to see (for example [18, Figure 9]) that: B n = n G a =1 r a . . . r n − B n − ⊔ n G a =1 t a r a . . . r n − B n − . So, if we define, for i ∈ { , . . . , n } , R ( i ) := { t ǫa r a . . . r i − | a ∈ { , . . . , i } , ǫ ∈ { , }} ; then we have that { u n . . . u | u i ∈ R ( i ) } , (4.10)forms a complete set of pairwise distinct elements of B n . Moreover this set consists of reducedexpressions in terms of the generators r , r , . . . , r n − , since the polynomial P k a k t k , where a k records the number of elements in (4.10) written as a product of k generators, is easilyfound to be Q ni =1 1 − t i − t which is the Poincaré polynomial P w ∈ B n t ℓ ( w ) of the Coxeter groupof type B n (see, for instance, [4, Theorem 7.1.5]).Now, to prove the lemma, we note that the subgroup permuting only the numbers ± , . . . , ± n is isomorphic to B (1) , the subgroup permuting only the numbers ± ( n +1) , . . . , ± ( n + n ) is isomorphic to B (2) and so on. These subgroups commute and thereforewe have an embedding of B (1) × · · · × B ( d ) inside B n (though not as a parabolic subgroup).It is straightforward to see that this corresponds to the embedding described at the level ofthe generators in the lemma.For the statement about the reduced expressions, let us first recall that the length func-tion of the Coxeter group B n can be expressed in terms on inversions as follows (see forexample [4, §8.1]): ℓ ( π ) = ♯ { ≤ i < j ≤ n | π ( i ) > π ( j ) } + ♯ { ≤ i ≤ j ≤ n | π ( − i ) > π ( j ) } . Using the notations of the lemma, we obtain that ℓ ( w . . . w d ) = ℓ ( w ) + · · · + ℓ ( w d ) ,since w permutes only the numbers ± , . . . , ± n , w permutes only the numbers ± ( n + ) , . . . , ± ( n + n ) , and so on. So it remains to show that a reduced expression in B ( j ) , j = 1 , . . . , d , is sent to a reduced expression in B n .Let j ∈ { , . . . , d } . We claim that it is enough to show our assertion for a single reducedexpression for each element of B ( j ) . Indeed the number of occurrences of r in differentreduced expressions of a same element remains constant (due to the homogeneity in r ofthe braid relations of B n ), and therefore, the number of generators in the images of thesedifferent reduced expressions is also constant. So if one of these images is reduced, they areall reduced.Finally, to conclude the proof of the lemma, we observe that the set of reduced expressionsof the form (4.10) in B ( j ) is sent to expressions of the same form in B n , which are thereforereduced as well.To prove that ρ is bijective, we use first that we know a basis of N dj =1 V β ( j ) (cid:0) Γ ( j ) , λ ( j ) , γ ( j ) (cid:1) by Theorem 3.26. A basis element is of the form d O j =1 ( y ( j )1 ) a ( j )1 . . . ( y ( j ) n j ) a ( j ) nj ψ ( j ) w j e ( i j ) , (4.11)where a ( j )1 , . . . a ( j ) n j ∈ N , i j ∈ β ( j ) and w j ∈ B ( j ) . Note that we have fixed a reducedexpression for each element w j ∈ B ( j ) for each j = 1 , . . . , d , in order to define ψ ( j ) w j .On the other hand, we also know a basis of e ( t β ) V β (Γ , λ, γ ) e ( t β ) again by Theorem 3.26.Indeed note that e ( i ) e ( t β ) = e ( i ) if the profile of i is t β and e ( i ) e ( t β ) = 0 otherwise.Moreover, ψ w e ( i ) = e ( w · i ) ψ w . So it is straightforward to conclude that a basis element of e ( t β ) V β (Γ , λ, γ ) e ( t β ) is of the form y a . . . y a n n ψ w e ( i ) , (4.12)where a , . . . , a n ∈ N , i ∈ β with profile t β and w is in the subgroup of B n isomorphic to B (1) × · · · × B ( d ) from Lemma 4.9 (the stabiliser of t β ). We must fix reduced expressionsfor such w in order to define ψ w . We fix them as the images of the reduced expressions ofelements B (1) × · · · × B ( d ) chosen in the preceding paragraph. That we can do so is the laststatement in Lemma 4.9.Finally, the image of a basis element (4.11) under the homomorphism ρ is y b . . . y b n n ψ w · · · ψ w d e ( i , . . . , i d ) , (4.13)where b n + ··· + n j − + k = a ( j ) k and the notation w j comes from Lemma 4.9. The concatenation ( i , . . . , i d ) has the profile t β since each i j ∈ β ( j ) , and due to our choices of reduced expres-sions, we have ψ w · · · ψ w d = ψ w ··· w d . So we conclude that the element (4.13) is of the form(4.12). Further, it is immediate that we can obtain in this way all the basis elements of e ( t β ) V β (Γ , λ, γ ) e ( t β ) . We conclude that the homomorphism ρ sends a basis onto a basis andthus is bijective. To finish the proof of Proposition 4.3, it remains to check that the map defined in (4.4)–(4.7)extends to an algebra homomorphism. It is possible but quite lengthy to check explicitly thatall defining relations are preserved. Instead we are going to use the polynomial representationintroduced in §3.2. We keep in use the notations introduced in §3.2.From the proof of Theorem 3.26, we see that the action of the algebra V β (Γ , λ, γ ) on K [ x, β ] is faithful, or in other words, we have an embedding of V β (Γ , λ, γ ) in End K ( K [ x, β ]) .Therefore, if we denote φ (cid:0) e ( t β ) (cid:1) the image of e ( t β ) by this embedding, we obtain an embed-ding of the algebra e ( t β ) V β (Γ , λ, γ ) e ( t β ) in End K (cid:0) φ (cid:0) e ( t β ) (cid:1) K [ x, β ] (cid:1) . We have immediately: φ (cid:0) e ( t β ) (cid:1) K [ x, β ] = M i ∈ βp ( i )= t β K [ x , . . . , x n ] i . (4.14) n the other hand, we also have an embedding of the algebra N dj =1 V β ( j ) (cid:0) Γ ( j ) , λ ( j ) , γ ( j ) (cid:1) in End K ( N dj =1 K [ x, β ( j ) ]) , and we have the natural identification: d O j =1 K [ x, β ( j ) ] = d O j =1 M i j ∈ β ( j ) K [ x ( j )1 , . . . , x ( j ) n j ] i j ∼ = M i ∈ βp ( i )= t β K [ x , . . . , x n ] i . (4.15)The identification simply maps f i ⊗ · · · ⊗ f d i d to f . . . f d ( i ,..., i d ) .Through the identifications we just made, both algebras related by the map in (4.4)–(4.7)are seen as algebras of endomorphisms of the same space, in (4.14) and (4.15). So in orderto check the homomorphism property, it is enough to check that both sides of Formulas(4.4)–(4.7) are in fact the same elements in the endomorphism algebra.This verification is immediate for (4.4)–(4.5) and (4.7). For the image of ψ ( j )0 , we proceedas follows. First, it is convenient to choose a polynomial representation as in §3.2 for which P ij ( u, v ) := ( u − v ) | j → i | if i = j and P ij ( u, v ) := 0 if i = j .Let i ∈ β such that p ( i ) = t β . It means that i = ( i , . . . , i d ) where i j ∈ β ( j ) . Fix j ∈ { , . . . , d } and set for brevity k = n + · · · + n j − . Through the identifications explainedabove, the action of ψ ( j )0 is given by: f i γ i k +1 f − r ( j )0 fx k +1 + α i k +1 ( x k +1 ) r ( j )0 f ! r ( j )0 · i , where we recall that r ( j )0 = r k . . . r r r . . . r k acts on i simply by replacing i k +1 by θ ( i k +1 ) .On the other hand, we need to calculate the action of ψ k . . . ψ ψ ψ . . . ψ k . We note that,with our choice of P ij ( u, v ) , we have that P ij ( u, v ) = 1 if one index is among { i , . . . , i k } and the other is i k +1 or θ ( i k +1 ) . Indeed, i k +1 is not in the same connected component ofthe quiver than i , . . . , i k since p ( i ) = t β . This is also true for θ ( i k +1 ) since θ leaves stablethe set I ( j ) .Then the calculation is made in three steps, corresponding respectively to the action of ψ . . . ψ k , the action of ψ and the action of ψ k . . . ψ : f i r ...r k f r ...r k · i (cid:18) γ i k +1 r ...r k f − r r ...r k fx + α i k +1 ( x ) r r ...r k f (cid:19) r r ...r k · i γ i k +1 f − r ( j )0 fx k +1 + α i k +1 ( x k +1 ) r ( j )0 f ! r ( j )0 · i . This concludes the verification of the homomorphism property and the proof of Proposi-tion 4.3. As in §2.2.4, let Λ = (Λ i ) i ∈ I be a finitely-supported family of non-negative integers. In thesame way as [25, 19, 20], we define the cyclotomic quotient of the algebra V β (Γ , λ, γ ) . Definition 4.16. We define the algebra V Λ β (Γ , λ, γ ) as the quotient of V β (Γ , λ, γ ) by thetwo-sided ideal J Λ β generated by the relations y Λ i e ( i ) = 0 , for all i = ( i , . . . , i n ) ∈ β .The above relations are homogeneous so that V Λ β (Γ , λ, γ ) is graded. Note that if Λ i = 0 for all i then V Λ β (Γ , λ, γ ) = ( { } , if n ≥ ,K, if n = 0 . s in §2.2.4, for any j ∈ { , . . . , d } let Λ ( j ) be the restriction of Λ to the vertex set I ( j ) of Γ ( j ) . Corollary 4.17. We have an (explicit) isomorphism of graded algebras: V Λ β (Γ , λ, γ ) ≃ Mat( nn ,...,nd ) d O j =1 V Λ ( j ) β ( j ) (cid:16) Γ ( j ) , λ ( j ) , γ ( j ) (cid:17) . Proof. Recall that the algebra R β (Γ) is isomorphic to a subalgebra of V β (Γ , λ, γ ) (see Corol-lary 3.28). Moreover, if ϑ denotes the isomorphism of Theorem 4.1, its restriction to R β (Γ) is by construction the isomorphism of Corollary 2.30. Therefore it is immediate that thecalculations made in the proof of Theorem 2.35 can be repeated verbatim here. Theyshow that, if we denote J Λ β, ⊗ the ideal of ⊗ dj =1 V β ( j ) (Γ ( j ) , λ ( j ) , γ ( j ) ) such that the quotient is ⊗ dj =1 V Λ ( j ) β ( j ) (Γ ( j ) , λ ( j ) , γ ( j ) ) (see the proof of Theorem 2.35), we have ϑ ( J Λ β ) = Mat( nn ,...,nd ) (cid:0) J Λ β, ⊗ (cid:1) . This concludes the proof.We define V Λ n (Γ , λ, γ ) := L β V Λ β (Γ , λ, γ ) where the direct sum is over the B n -orbits β in I n . As in Corollary 2.32, using the bijection (2.17) we deduce the following corollary.Note that we now use (2.17) with G = Z / Z . Corollary 4.18. We have an (explicit) isomorphism of graded algebras: V Λ n (Γ , λ, γ ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) d O j =1 V Λ ( j ) n j (Γ ( j ) , λ ( j ) , γ ( j ) ) . Remark . As in Remark 2.39, we deduce that we can assume that Λ is supported on allcomponents of Γ . To fit with the setting of [20], we now assume that K is a field with char ( K ) = 2 .Let Γ be a quiver with an involution θ as in §3.1 and let β be a B n -orbit in I n . As before,let Λ = (Λ i ) i ∈ I be a finitely-supported family of non-negative integers.In this section, as in Remark 3.18 we consider the situation λ i = γ i = 0 for all i ∈ I , andwe denote simply V β (Γ) = V β (Γ , , the resulting algebra, defined in Section 3.1 (note thatConditions (3.5) are satisfied with this choice of λ and γ ). The defining relations (3.9)–(3.14)(those involving the generator ψ ) become simply: ψ e ( i ) = e ( r · i ) ψ , (5.1) ψ ψ b = ψ b ψ , for all b ∈ { , . . . , n − } , (5.2) ψ y = − y ψ , (5.3) ψ y a = y a ψ , for all a ∈ { , . . . , n } , (5.4) ψ = 1 , (5.5) ( ψ ψ ) = ( ψ ψ ) . (5.6)So we see immediately that we have an homogeneous involutive algebra automorphism ι of V β (Γ) given on the generators by: ι ( ψ ) = − ψ and ι ( X ) = X for X ∈ { ψ , . . . , ψ n − , y , . . . , y n } ∪ { e ( i ) } i ∈ β . (5.7)Note that ι is the identity map if n = 0 . We denote by V β (Γ) ι the fixed-point subalgebraof V β (Γ) , that is, V β (Γ) ι = { x ∈ V β (Γ) | ι ( x ) = x } . The subalgebra V β (Γ) ι is a gradedsubalgebra of V β (Γ) since ι is homogeneous. yclotomic quotients. We recall that V Λ β (Γ) is the quotient of V β (Γ) by the two-sidedideal J Λ β generated by y Λ i e ( i ) = 0 , for all i ∈ β .These relations are homogeneous so that the algebra V Λ β (Γ) inherits the grading of V β (Γ) .The same formulas as in (5.7) define an homogeneous involutive algebra automorphism of V Λ β (Γ) , and we make the slight abuse of notation of keeping the name ι for this automorphism.The fixed-point subalgebra is denoted V Λ β (Γ) ι . W δ (Γ) We recall some definitions and the results we need from [20].If n ≥ , we identify the Weyl group D n of type D as the subgroup of B n generated by s := r r r , s := r , . . . , s n − := r n − . The convention we need here is that D n = { } if n ∈ { , } . The group D n then acts on I n by, if n ≥ , s · ( i , i , . . . , i n ) = ( θ ( i ) , θ ( i ) , i , . . . , i n ) ,s a · ( . . . , i a , i a +1 , . . . ) = ( . . . , i a +1 , i a , . . . ) a = 1 , . . . , n − . Let δ be a finite subset of I n stable by D n , that is a finite union of D n -orbits. Definition 5.8. Let n ≥ . The algebra W δ (Γ) is the unitary associative K -algebra gener-ated by elements { y a } ≤ a ≤ n ∪ { ψ b } ≤ b ≤ n − ∪ { Ψ } ∪ { e ( i ) } i ∈ δ , with the relations (2.5)–(2.10) of Section 2 involving all the generators but Ψ , together with Ψ e ( i ) = e ( s · i )Ψ , (5.9) Ψ ψ b = ψ b Ψ , for all b ∈ { , . . . , n − } with b = 2 , (5.10) (Ψ y a + y r ( a ) Ψ ) e ( i ) = (cid:26) e ( i ) if θ ( i ) = i , otherwise, for a ∈ { , } , (5.11) Ψ y a = y a Ψ , for all a ∈ { , . . . , n } , (5.12) Ψ e ( i ) = Q θ ( i ) ,i ( − y , y ) e ( i ) , (5.13) (Ψ ψ Ψ − ψ Ψ ψ ) e ( i ) = Q θ ( i ) ,i ( − y , y ) − Q θ ( i ) ,i ( y , y ) y + y e ( i ) , if θ ( i ) = i , , otherwise , (5.14)for all i ∈ δ .By convention, we set W δ (Γ) = R δ (Γ) if n ∈ { , } . Explicitly, W δ (Γ) = K if n = 0 and W δ (Γ) = P i ∈ δ K [ y ] e ( i ) if n = 1 . This choice for n ∈ { , } is important for the statementsof the results in the next subsection. Remark . With the choices of Γ , θ and the notations of Remark 3.17, the algebra W δ (Γ) is exactly the algebra W δ x defined in [20].The algebra W δ (Γ) is Z -graded with deg e ( i ) = 0 , deg y a = 2 , deg Ψ e ( i ) = d ( θ ( i ) , i ) , deg ψ b e ( i ) = d ( i b , i b +1 ) . Definition 5.16. The cyclotomic quotient W Λ δ (Γ) is the quotient of the algebra W δ (Γ) bythe relations y Λ i e ( i ) = 0 , for all i ∈ δ . he algebra W Λ δ (Γ) inherits the grading from W δ (Γ) since the additional relations arehomogeneous. If Λ i = 0 for all i then W Λ δ (Γ) = ( { } , if n ≥ ,K, if n = 0 . Fixed-point isomorphism. Let β be a B n -orbit in I n . Note that β is a finite unionof D n -orbits, so that both algebras V β (Γ) and W β (Γ) are defined.We recall the following results from [20]. Note that they were proved for a particularchoice of Γ and θ (the one relevant for the next section). However, the proof does not dependon this choice and can be repeated verbatim in our general setting. Proposition 5.17 ([20]) . (i) The algebra W β (Γ) is isomorphic to the subalgebra V β (Γ) ι of V β (Γ) .(ii) Assume that Λ satisfies Λ θ ( i ) = Λ i for all i ∈ I . The cyclotomic quotient W Λ β (Γ) isisomorphic to V Λ β (Γ) ι .In both cases, an isomorphism is given by Ψ ψ ψ ψ and X X for all the genera-tors X but Ψ .Remark . Note that it is assumed in [20] that n ≥ . With our conventions, thestatements are also true for n ∈ { , } , in which cases the verification is straightforward. Remark . Recall the defining relations (5.1), (5.3) and (5.5) of V Λ β (Γ) . Conjugating thecyclotomic relations of V Λ β (Γ) by ψ , we obtain that y Λ i e ( r · i ) = 0 for any i ∈ β . Fromthis remark, it is easy to see that we have in fact V Λ β (Γ) = V e Λ β (Γ) , where e Λ is now given by e Λ i = min { Λ i , Λ θ ( i ) } . This phenomenon does not necessarily occur also in W Λ β (Γ) (where ψ is not present) and this explains the assumptions on Λ in Proposition 5.17(ii).We note that the isomorphisms given in the preceding proposition are isomorphisms ofgraded algebras. Indeed, in V β (Γ) we have deg ψ = 0 and so it is straightforward to checkthat the given map is homogeneous.From Proposition 5.17(i) and Corollary 3.28, one obtains immediately the following state-ment. Corollary 5.20. The subalgebra of W β (Γ) generated by all generators but Ψ is isomorphicto R β (Γ) . Semi-direct product. In this paragraph, assume that n ≥ . Since ι is involutive, thevector space V β (Γ) decomposes as V β (Γ) = V β (Γ) ι ⊕ V β (Γ) − , where V β (Γ) − is the eigenspace of ι for the eigenvalue − . Moreover, the generator ψ isinvertible (in fact, ψ = 1 ) and satisfies ι ( ψ ) = − ψ . So the multiplication by ψ providesan isomorphism of vector spaces between V β (Γ) ι and V β (Γ) − , so that V β (Γ) − can be written V β (Γ) ι ψ . Working out the multiplication in V β (Γ)( x + yψ )( x ′ + y ′ ψ ) = xx ′ + yψ y ′ ψ + ( yψ x ′ ψ + xy ′ ) ψ , one obtains as a standard consequence that V β (Γ) is isomorphic to the semi-direct product V β (Γ) ι ⋊ C , where the action of the cyclic group C of order 2 on V β (Γ) ι is by conjugationby ψ . Recall that as a vector space V β (Γ) ι ⋊ C is the tensor product V β (Γ) ι ⊗ K [ C ] , andthe multiplication is given by ( x ⊗ ψ ǫ )( x ′ ⊗ ψ ǫ ′ ) = ( xψ ǫ x ′ ψ ǫ ) ⊗ ψ ǫ + ǫ ′ . Then we formulate the preceding standard facts taking into account Proposition 5.17.First we give explicitly the automorphism of W β (Γ) induced by conjugation by ψ in V β (Γ) .We denote this automorphism of order 2 by π . It is given on the generators by: π : Ψ ψ , ψ Ψ , y 7→ − y , e ( i ) e ( r · i ) , (5.21) nd the identity on all the other generators. As a consequence of Proposition 5.17 togetherwith the preceding discussion, we conclude that V β (Γ) ≃ W β (Γ) ⋊ h π i , and similarly, for Λ as in Proposition 5.17(ii), V Λ β (Γ) ≃ W Λ β (Γ) ⋊ h π i , (5.22)where we still denote by π the automorphism of order 2 of W Λ β (Γ) given by the sameformulas (5.21). This is indeed an automorphism since Λ satisfies the assumption of Propo-sition 5.17(ii).With these descriptions as semi-direct products, the involution ι on V β (Γ) (and on V Λ β (Γ) )is simply given by: ι (cid:0) x ⊗ π ǫ (cid:1) = ( − ǫ x ⊗ π ǫ , (5.23)where ǫ ∈ { , } and x ∈ W β (Γ) (or x ∈ W Λ β (Γ) ). W δ (Γ) Now let d be a positive integer and assume that the quiver Γ admits a decomposition Γ = ∐ dj =1 Γ ( j ) as in §4. Let β be a B n -orbit in I n . As in §4, for any j ∈ { , . . . , d } , we havean integer n j ( β ) = n j and a B n j -orbit β ( j ) in ( I ( j ) ) n j .If n j ( β ) = 0 for some j ∈ { , . . . , d } then consider ˜Γ the quiver where we removed thecomponent Γ ( j ) . It is immediate from the definitions that W β (Γ) is the same algebra as W β (˜Γ) . So we lose no generality by assuming that n j ( β ) = 0 for all j ∈ { , . . . , d } . Fixed points of tensor products. Since n j ( β ) ≥ for all j ∈ { , . . . , d } , by thepreceding section we have V β ( j ) (Γ ( j ) ) ≃ W β ( j ) (Γ ( j ) ) ⋊ C for all j . Hence, d O j =1 V β ( j ) (Γ ( j ) ) ≃ (cid:16) d O j =1 W β ( j ) (Γ ( j ) ) (cid:17) ⋊ C d , where C d acts on the tensor product by the automorphism π from (5.21) on each factor.We would like to describe the fixed points of N dj =1 V β ( j ) (Γ ( j ) ) for the involutive auto-morphism ι ⊗ given by the tensor product of ι for each factor. From Formula (5.23), it isimmediate to see that (cid:16) d O j =1 V β ( j ) (Γ ( j ) ) (cid:17) ι ⊗ ≃ (cid:16) d O j =1 W β ( j ) (Γ ( j ) ) (cid:17) ⋊ C d − , (5.24)where C d − is seen as the subgroup of “even” elements of C d , namely C d − = { ( π ǫ , . . . , π ǫ d ) ∈ C d such that ǫ + · · · + ǫ d = 0 (mod 2) } . (5.25) Disjoint quiver isomorphism. We can now formulate the main result of this section.Recall that n j ( β ) = 0 for all j ∈ { , . . . , d } . Theorem 5.26. We have (explicit) isomorphisms of graded algebras: W β (Γ) ≃ Mat( nn ,...,nd ) (cid:16) d O j =1 W β ( j ) (Γ ( j ) ) (cid:17) ⋊ C d − , (5.27) and, assuming d > , W Λ β (Γ) ≃ Mat( nn ,...,nd ) (cid:16) d O j =1 W e Λ ( j ) β ( j ) (Γ ( j ) ) (cid:17) ⋊ C d − , (5.28) where e Λ = ( e Λ i ) i ∈ I is defined by e Λ i := min { Λ i , Λ θ ( i ) } . ote that in both formulas above, the group C d − is as given in (5.25). Moreover, thesemi-direct product in Formula (5.28) is well-defined since each e Λ ( j ) satisfies the condition e Λ ( j ) i = e Λ ( j ) θ ( i ) of Proposition 5.17(ii) (see (5.22)). Remark . The reader may have noticed that the assumptions d > and n j ( β ) = 0 (which do not reduce the generality as explained above) were not present in the precedingsection for the type B in Theorem 4.1 and Corollary 4.17. Indeed those statements are moreuniform in the sense that they are also valid as they are, even if some n j ( β ) are 0 or if d = 1 .In particular, for d = 1 we do not necessarily have W Λ β (Γ) = W e Λ β (Γ) (cf. Remark 5.19). Proof. • Recall from Theorem 4.1 that we have an isomorphism between V β (Γ) and thealgebra Mat( nn ,...,nd ) (cid:16)N dj =1 V β ( j ) (Γ ( j ) ) (cid:17) . This isomorphism was obtained with the followingtwo steps: V β (Γ) ≃ Mat( nn ,...,nd ) (cid:0) e ( t β ) V β (Γ) e ( t β ) (cid:1) and d O j =1 V β ( j ) (Γ ( j ) ) ≃ e ( t β ) V β (Γ) e ( t β ) . For the first isomorphism, see §4.1, the construction of the idempotent e ( t β ) does not in-volve ψ , and neither does the construction of the matrix units (that is, the constructionof the elements ψ t and φ t given by Formulas (2.21)). So we deduce immediately how theautomorphism ι of V β (Γ) behaves with respect to this isomorphism, namely we have that V β (Γ) ι ≃ Mat( nn ,...,nd ) (cid:0) e ( t β ) V β (Γ) ι e ( t β ) (cid:1) . According to Formula (5.24) (that we can use since n j ( β ) = 0 ), to prove (5.27) it remainsonly to show that e ( t β ) V β (Γ) ι e ( t β ) ≃ (cid:16) d O j =1 V β ( j ) (Γ ( j ) ) (cid:17) ι ⊗ . So if we denote ρ the isomorphic map from N dj =1 V β ( j ) (Γ ( j ) ) to e ( t β ) V β (Γ) e ( t β ) , it remainsto check that ρ ◦ ι ⊗ = ι ◦ ρ . This is immediately verified from Formulas (4.4)–(4.7) giving the map ρ in the proof ofProposition 4.3. Moreover, the isomorphism (5.27) is graded since it is the restriction of agraded isomorphism (to a graded subalgebra). • To prove (5.28), we start exactly as in the proof of Corollary 4.17, namely we repeatthe calculations in the proof of Theorem 2.35. We can do so since R β (Γ) is a subalgebra of W β (Γ) by Corollary 5.20.Let ϑ denote the isomorphism in (5.27) and let K Λ β denote the ideal of W β (Γ) giving thecyclotomic quotient W Λ β (Γ) . The proofs of Corollary 4.17 and Theorem 2.35 show that ϑ ( K Λ β ) = Mat( nn ,...,nd )( K Λ β, ⊗ ) , where K Λ β, ⊗ is the ideal of (cid:16)N dj =1 W β ( j ) (Γ ( j ) ) (cid:17) ⋊ C d − generated by the elements y Λ ib b e ( i ) , (5.30)where i is of profile t β , and b is of the form b = n + · · · + n j − + 1 for j ∈ { , . . . , d } . Notethat, as in the proof of Theorem 2.35 we slightly abuse notations: if i = ( i , . . . , i d ) with i k ∈ β ( k ) , we identify y Λ ib b e ( i ) ∈ W β (Γ) with the element of N dj =1 W β ( j ) (Γ ( j ) ) which is e ( i k ) in the k -th factor with k = j and ( y ( j )1 ) Λ ( i j )1 e ( i j ) in the j -th factor (where y ( j )1 denotes thegenerator y of W β ( j ) (Γ ( j ) ) ). ontrary to the type A and B, we need to show something more here to prove (5.28).In particular, we cannot consider the semi-direct product (cid:16)N dj =1 W Λ ( j ) β ( j ) (Γ ( j ) ) (cid:17) ⋊ C d − sincethe elements Λ ( j ) do not necessarily satisfy the stability condition of Proposition 5.17(ii).Thus, let K e Λ β, ⊗ be the ideal of (cid:16)N dj =1 W β ( j ) (Γ ( j ) ) (cid:17) ⋊ C d − generated by the elements y e Λ ib b e ( i ) , (5.31)where i ∈ β is of profile t β , and b is of the form b = n + · · · + n j − + 1 for j ∈ { , . . . , d } ,and where e Λ is defined in Theorem 5.26. We will show that K Λ β, ⊗ = K e Λ β, ⊗ . First, since e Λ i ≤ Λ i for all i ∈ I , we have K Λ β, ⊗ ⊂ K e Λ β, ⊗ . For the reverse inclusion, takean element y e Λ ib b e ( i ) as in (5.30). If e Λ i b = Λ i b then y e Λ ib b e ( i ) ∈ K Λ β, ⊗ , thus we assume that e Λ i b = Λ θ ( i b ) . Let ξ ∈ C d − such that the component of ξ in position j is π . Such an elementexists since we assumed that d > . Then, using Formulas (5.21) for the action of π on W β ( j ) (Γ ( j ) ) , we have, where i ′ ∈ β of profile t β is such that i ′ b = θ ( i b ) , ξ · (cid:18) y e Λ ib b e ( i ) (cid:19) = ( − y b ) e Λ ib e ( i ′ ) = ( − y b ) Λ θ ( ib ) e ( i ′ ) = ( − y b ) Λ i ′ b e ( i ′ ) . Since the action of ξ is invertible, we thus deduce that y e Λ ib b e ( i ) ∈ K Λ β, ⊗ . Finally, we showedthat all elements in (5.31) are in K Λ β, ⊗ , and thus K e Λ β, ⊗ ⊂ K Λ β, ⊗ . This concludes the proof.We define W n (Γ) := ⊕ δ W δ (Γ) , where δ runs over all the orbits of I n under the actionof D n , and similarly W Λ n (Γ) = ⊕ δ W Λ δ (Γ) . In the type D situation, the statements beloware less clean that those of Corollary 2.32 or Corollary 4.18. Nevertheless, it still explicitlyreduces the study of W n (Γ) and W Λ n (Γ) to the situation of a quiver with a single component.For ( n , . . . , n d ) ∈ ( Z ≥ ) d , we denote l ( n , . . . , n d ) the number of its non-zero components.Assume that n ≥ to avoid a trivial situation. Corollary 5.32. We have (explicit) isomorphisms of graded algebras: W n (Γ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) (cid:16) d O j =1 n j =0 W n j (Γ ( j ) ) (cid:17) ⋊ C l ( n ,...,n d ) − ,W Λ n (Γ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) (cid:16) W ( n , . . . , n d ) (cid:17) , where: • If l ( n , . . . , n d ) = 1 then W ( n , . . . , n d ) := W Λ ( j ) n j (Γ ( j ) ) where j is the component suchthat n j = n . • If l ( n , . . . , n d ) > then W ( n , . . . , n d ) := (cid:16) d O j =1 n j =0 W e Λ ( j ) n j (Γ ( j ) ) (cid:17) ⋊ C l ( n ,...,n d ) − . Proof. We write W n (Γ) = ⊕ β W β (Γ) and W Λ n (Γ) = ⊕ β W Λ β (Γ) , where β runs over all theorbits of I n under the action of B n . We note that if some n j ( β ) are equal to 0 then, asexplained at the beginning of this subsection, we can remove the corresponding componentsof Γ to obtain another quiver ˜Γ for which the assumptions of Theorem 5.26 are satisfied.Then the proof is a repetition of the proof of Corollary 2.32, using Theorem 5.26 for eachorbit β . emark . As in Remarks 2.39 and 4.19, we deduce that we can assume that Λ is sup-ported on all the components of Γ . In this section, we will combine our previous results Corollaries 4.18 and 5.32 with [19, 20]to obtain Morita equivalences theorems for cyclotomic quotients of affine Hecke algebras oftype B and D. We emphasize that these Morita equivalences will be deduced from isomor-phisms. As they combine the isomorphisms of [19, 20] with those of the previous sections,these isomorphisms can be written down explicitly even though they are rather complicated.Recall that K is a field with characteristic different from two. Let p, q ∈ K \ { } suchthat q = 1 . As in Remark 3.17, for any x ∈ K \ { } we define the set I x := { x ǫ q l : ǫ ∈ {± } , l ∈ Z } . Then we take d ≥ and x , . . . , x d ∈ K × such that the sets I ( j ) := I x j are pairwise disjoint,and we set I := ∐ dj =1 I x j . The quiver Γ with involution that we will be considering in this section is the following:• The vertex set of Γ is I as above.• There is an arrow starting from v and pointing to q v for all v ∈ I . These are all arrows.• The involution θ on I is the scalar inversion θ ( x ) = x − for all x ∈ I .The partition I = ∐ dj =1 I ( j ) induces a decomposition of Γ into full subquivers Γ = ∐ dj =1 Γ ( j ) as in Section 4, in particular each Γ ( j ) is stable under the scalar inversion θ . Wealso choose a finitely-supported family Λ = (Λ i ) i ∈ I of non-negative integers. Finally, let L be a free Z -module of rank n with basis { ǫ i } i =1 ,...,n : L := n M i =1 Z ǫ i . We set α := 2 ǫ and α i := ǫ i +1 − ǫ i , i = 1 , . . . , n − . For n ≥ , the Weyl group B n of type B acts on L by r ( ǫ ) = − ǫ ,r ( ǫ i ) = ǫ i if i > ,r a ( ǫ i ) = ǫ r a ( i ) , for i = 1 , . . . , n − and a = 1 , . . . , n − .We denote q := p and q i := q for i = 1 , . . . , n − . The affine Hecke algebra b H ( B n ) isthe unitary K -algebra generated by elements g , g , . . . , g n − and X x , x ∈ L . The defining relations are X = 1 , X x X x ′ = X x + x ′ for any x, x ′ ∈ L , and the characteristicequations for the generators g i : g i = ( q i − q − i ) g i + 1 for i ∈ { , . . . , n − } , (6.1) ith the braid relations of type B g g g g = g g g g (6.2) g i g i +1 g i = g i +1 g i g i +1 for i ∈ { , . . . , n − } , (6.3) g i g j = g j g i for i, j ∈ { , . . . , n − } such that | i − j | > , (6.4)together with g i X x − X r i ( x ) g i = ( q i − q − i ) X x − X r i ( x ) − X − α i , for any x ∈ L and i = 0 , , . . . , n − . Note that the right-hand side is a well-defined elementsince there exists k ∈ Z such that r i ( x ) = x − kα i . Note also that b H ( B ) = K .Let X i := X ǫ i for i = 1 , . . . , n . An equivalent presentation of the algebra b H ( B n ) is withgenerators g , g , . . . , g n − , X ± , . . . , X ± n , and defining relations (6.1)–(6.4) together with X i X j = X j X i for i, j ∈ { , . . . , n } ,g X − g = X ,g i X i g i = X i +1 for i ∈ { , . . . , n − } ,g i X j = X j g i for i ∈ { , . . . , n − } and j ∈ { , . . . , n } such that j = i, i + 1 . Definition 6.5. The cyclotomic quotient H Λ ( B n ) of type B associated with Λ = (Λ i ) i ∈ I isthe quotient of the algebra b H ( B n ) over the relation Y i ∈ I ( X − i ) Λ i = 0 . Note that if Λ i = 0 for all i then H Λ ( B n ) = ( { } , if n ≥ ,K, if n = 0 . We recall the main result of [19, 20] concerning H Λ ( B n ) . Theorem 6.6. Let λ, γ be as in Remarks 3.17 and 3.18 if p = 1 and p = 1 respectively.The algebras H Λ ( B n ) and V Λ n (Γ , λ, γ ) are (explicitly) isomorphic.Remark . Theorem 6.6 is proven for n ≥ , but is also trivially true for n = 0 .We now state the first main application of the results of the preceding sections. Theorem 6.8. We have an (explicit) isomorphism of algebras: H Λ ( B n ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) d O j =1 H Λ ( j ) ( B n j ) . In particular, H Λ ( B n ) is Morita equivalent to M n ,...,n d ≥ n + ··· + n d = n d O j =1 H Λ ( j ) ( B n j ) .Proof. Note that the statement is true if n = 0 , thus we now assume n ≥ . Let us firstassume that p = 1 . Let λ be the indicator function of {± p } ∩ I and ( γ i ) i ∈ I be givenby γ i = ( , if θ ( i ) = i, , otherwise , as in Remark 3.17. By Theorem 6.6, we have an isomorphism Λ ( B n ) ≃ V Λ n (Γ , λ, γ ) . For any j ∈ { , . . . , d } , the restrictions λ ( j ) and γ ( j ) of λ and γ respectively to I ( j ) satisfy, by Corollary 4.18, V Λ n (Γ , λ, γ ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) d O j =1 V Λ ( j ) n j (Γ ( j ) , λ ( j ) , γ ( j ) ) . Since λ ( j ) and γ ( j ) are still of the above form with respect to the quiver Γ ( j ) , by Theorem 6.6we have V Λ ( j ) n j (Γ ( j ) , λ ( j ) , γ ( j ) ) ≃ H Λ ( j ) ( B n j ) for any n j . We thus deduce the isomorphismof the theorem. We deduce the statement of Morita equivalence since Mat N ( A ) and A areMorita equivalent for any algebra A and N ∈ N ∗ . The case p = 1 is similar, still byTheorem 6.6.We obtain the following corollary. Corollary 6.9. To study an arbitrary cyclotomic quotient of the affine Hecke algebra b H ( B n ) ,it is enough to consider cyclotomic quotients given by a relation Y ǫ ∈{± } l ∈ Z ( X − x ǫ q l ) m ǫ,l = 0 , for any finitely-supported family of non-negative integers ( m ǫ,l ) ǫ ∈{± } ,l ∈ Z , where x ∈ K × satisfies one of the following four cases: ( a ) x = 1 ( b ) x = q ( c ) x = p ( d ) x / ∈ ± q Z ∪ ± p ± q Z . Proof. We sketch a proof, in the same spirit as in the introduction of [19]. By Theorem 6.8,it is clear that it suffices to consider cyclotomic quotients given by a relation Y i ∈ I x ( X − i ) Λ i = 0 , where I x = { x ǫ q l : ǫ ∈ {± } , l ∈ Z } with x ∈ K × and Λ = (Λ i ) i ∈ I x is a finitely-supportedfamily of non-negative integers. By Theorem 6.6 and Remark 3.17, this cyclotomic quotientis determined by:• the quiver Γ with vertex set I x , arrows v → q v for all v ∈ I x and involution θ : v v − on I x ;• the set {± p } ∩ I x .A first distinction arises when looking at the number of connected components of Γ . Ithas exactly one (respectively two) connected component(s) when x ∈ q Z (resp. x / ∈ q Z ).The first case, x ∈ q Z , is equivalent to x ∈ ± q Z . We can switch between x and − x by the variable change X i ← − X i for all i ∈ { , . . . , n } , replacing I x by − I x = I − x and Λ = (Λ i ) i ∈ I x by Λ ′ = (Λ ′ i ) i ∈ I − x given by Λ ′ i := Λ − i for all i ∈ I − x . Thus, it suffices toconsider x ∈ q Z , but now a simple shift of Λ (that is, setting Λ ′ i = Λ iq N for appropriate N )shows that it suffices to consider the cases x = 1 (this is case ( a ) ) or x = q (this is case ( b ) ),according to the parity of the power of q .We now consider the case x / ∈ q Z , that is, x / ∈ ± q Z . If {± p } ∩ I x = ∅ , then x / ∈± q Z ∪ ± p ± q Z , and all these choices of x lead to isomorphic algebras since moreover θ hasno fixed points (if x ± q k is fixed by θ then x ∈ q Z thus x ∈ ± q Z ). This is case ( d ) . Nowif {± p } ∩ I x = ∅ , using the variable change X i ← − X i for all i ∈ { , . . . , n } we can alwaysassume that p ∈ I x , that is, x ∈ p ± q Z . It suffices in fact to consider x ∈ pq Z , since thevariable change g ← − g exchanges p and p − . This case reduces to x = p by shifting Λ asabove, and this is case ( c ) . Remark . We make additional final remarks on the four cases ( a ) – ( d ) to be considered. Cases ( a ) and ( b ) correspond to a quiver with a single connected component (an infiniteoriented line or a finite oriented polygon depending on whether q is a root of unity ornot). This quiver is stable by the involution θ , and then Case ( a ) corresponds to θ having a fixed point, while Case ( b ) generically corresponds to the situation wherethere is no fixed point. This latter situation cannot occur if the number of vertices isfinite and odd, that is, Case ( b ) is not present (or more precisely, is not necessary sinceit is equivalent to Case ( a ) ) when q is an odd root of unity.• Cases ( c ) and ( d ) (generically) correspond to a quiver with two identical connectedcomponents (two infinite oriented lines or two finite oriented polygons depending onwhether q is a root of unity or not), which are exchanged by the involution θ . ThenCase ( c ) corresponds to the situation where one of the special values ± p − is present,while Case ( d ) corresponds to the situation where no such values occur. We see thatCase ( c ) is not necessary (more precisely, it reduces to one of Cases ( a ) or ( b ) ) when p is a power of q .• To summarise, there are at least two cases to consider in general: ( a ) and ( d ) , whilethe additional two cases ( b ) and ( c ) are to be considered or not depending on p and q . Let n ≥ . We set α ′ = ǫ + ǫ and α ′ i = ǫ i +1 − ǫ i , i = 1 , . . . , n − . The Weyl group D n of type D acts on L by s ( ǫ ) = − ǫ ,s ( ǫ ) = − ǫ ,s ( ǫ i ) = ǫ i , if i > ,s a ( ǫ i ) = ǫ r a ( i ) , for i = 1 , . . . , n − and a = 1 , . . . , n − .The affine Hecke algebra b H ( D n ) is the unitary K -algebra generated by elements { g i } ≤ i ≤ n − ∪ { G } ∪ { X x } x ∈ L . The defining relations are X = 1 , X x X x ′ = X x + x ′ for any x, x ′ ∈ L , and the characteristicequations for the generators g i and G : g i = ( q − q − ) g i + 1 for i ∈ { , . . . , n − } ,G = ( q − q − ) G + 1 , (6.11)with the braid relations of type D G g G = g G g , (6.12) G g i = g i G for i ∈ { , . . . , n − } \ { } , (6.13) g i g i +1 g i = g i +1 g i g i +1 for i ∈ { , . . . , n − } , (6.14) g i g j = g j g i for i, j ∈ { , . . . , n − } such that | i − j | > , (6.15)together with g i X x − X s i ( x ) g i = ( q − q − ) X x − X s i ( x ) − X − α ′ i ,G X x − X s ( x ) G = ( q − q − ) X x − X s ( x ) − X − α ′ , or any x ∈ L and i = 1 , . . . , n − . Note that the right-hand sides are well-defined elementssince for any i ∈ { , . . . , n − } there exists k ∈ Z such that s i ( x ) = x − kα ′ i .An equivalent presentation of the algebra b H ( D n ) is with generators (where again X i := X ǫ i ) { g i } ≤ i ≤ n − ∪ { G } ∪ { X ± i } ≤ i ≤ n , and defining relations (6.11)–(6.15) together with X i X j = X j X i for i, j ∈ { , . . . , n } ,G X − G = X ,G X i = X i G for i ∈ { , . . . , n − } ,g i X i g i = X i +1 for i ∈ { , . . . , n − } ,g i X j = X j g i for i ∈ { , . . . , n − } and j ∈ { , . . . , n } such that j = i, i + 1 . By convention, we set that b H ( D n ) coincides with the usual affine Hecke algebra of type A n if n ∈ { , } , that is, we have b H ( D ) = K and b H ( D ) = K [ X ± ] . Definition 6.16. The cyclotomic quotient H Λ ( D n ) of type D associated with Λ = (Λ i ) i ∈ I is the quotient of the algebra b H ( D n ) over the relation Y i ∈ I ( X − i ) Λ i = 0 . Note that if Λ i = 0 for all i then H Λ ( D n ) = ( { } , if n ≥ ,K, if n = 0 . We recall the main result of [20] concerning H Λ ( D n ) . Recall that the quiver Γ wasdefined at the beginning of Section 6. Theorem 6.17. The algebras H Λ ( D n ) and W Λ n (Γ) are (explicitly) isomorphic.Remark . Theorem 6.17 is proven for n ≥ , but it is immediate with our conventionsthat it remains true for n ∈ { , } . Expression as a semi-direct product. We assume here that n ≥ . Assuming p = 1 , we now can see b H ( D n ) as a subalgebra of b H ( B n ) . Namely, we have an inclusion(see, for instance, [20, §2.3]) b H ( D n ) ⊆ b H ( B n ) , given on the generators by G g g g , g i g i , X ± j X ± j , for any i ∈ { , . . . , n − } and j ∈ { , . . . , n } . Another way to see b H ( D n ) as a subalgebra of b H ( B n ) is to write b H ( D n ) as the subalgebra of fixed points of b H ( B n ) under the involution η given by g 7→ − g , g i g i , X ± j X ± j , for each i ∈ { , . . . , n − } and j ∈ { , . . . , n } (note that since p = 1 the defining relation forthe generator g is g = 1 ). In particular, as in §5.1 we have a vector space decomposition b H ( B n ) = b H ( D n ) ⊕ b H ( D n ) g and thus an isomorphism of algebras b H ( B n ) ≃ b H ( D n ) ⋊ C . Note that the action of the generator of C on the generating set of b H ( D n ) is given by G g , g G , g i g i ,X X − , X − X , X ± j X ± j , or all i ∈ { , . . . , n − } and j ∈ { , . . . , n } .The involution η on b H ( B n ) is compatible with the cyclotomic quotient H Λ ( B n ) . Nowif Λ satisfies the stability condition of Proposition 5.17(ii) (which is here Λ i − = Λ i for all i ∈ I ), the previous action of C on b H ( D n ) is compatible with the cyclotomic quotient H Λ ( D n ) and as above we have H Λ ( B n ) ≃ H Λ ( D n ) ⋊ C . Morita equivalence theorem. Let n , . . . , n d ≥ . If Λ satisfies Λ i − = Λ i forall i ∈ I , the previous action of C on H Λ ( D n ) extends to a (diagonal) action of C d on ⊗ dj =1 H Λ ( j ) ( D n j ) . As in §5.2, we restrict this action to the subgroup C d − of even elementsgiven in (5.25). Recall also the definition of e Λ = ( e Λ i ) i ∈ I given in Theorem 5.26.We now state the second main application of the paper. As in Corollary 5.32, for any ( n , . . . , n d ) ∈ ( Z ≥ ) d we denote by l ( n , . . . , n d ) the number of its non-zero components. Theorem 6.19. We have an (explicit) isomorphism of algebras: H Λ ( D n ) ≃ M n ,...,n d ≥ n + ··· + n d = n Mat( nn ,...,nd ) (cid:16) H ( n , . . . , n d ) (cid:17) , where: • If l ( n , . . . , n d ) = 1 then H ( n , . . . , n d ) := H Λ ( j ) ( D n j ) where j is the component suchthat n j = n . • If l ( n , . . . , n d ) > then H ( n , . . . , n d ) := (cid:16) d O j =1 n j =0 H e Λ ( j ) ( D n j ) (cid:17) ⋊ C l ( n ,...,n d ) − . In particular, H Λ ( D n ) is Morita equivalent to M n ,...,n d ≥ n + ··· + n d = n H ( n , . . . , n d ) .Proof. We argue as in the proof of Theorem 6.8, using Corollary 5.32 and Theorem 6.17.Note that the isomorphism of [20] is compatible with the semi-direct product since theinvolution ι (respectively, the element ψ ) of V Λ n (Γ) is sent to the involution η (resp., theelement g ) of H Λ ( B n ) by the isomorphism of loc. cit. We obtain the following corollary. We note that the situation is a little bit more intricatethan for type B because of the presence of semidirect products with products of groups C .So below, it is implicit that it is enough to consider some special cyclotomic quotients, upto the application of standard Clifford theory to deal with the semidirect products. Corollary 6.20. To study an arbitrary cyclotomic quotient of the affine Hecke algebra b H ( D n ) , it is enough to consider cyclotomic quotients given by a relation Y ǫ ∈{± } l ∈ Z ( X − x ǫ q l ) m ǫ,l = 0 , for any finitely-supported family of non-negative integers ( m ǫ,l ) ǫ ∈{± } ,l ∈ Z , where x ∈ K × satisfies one of the following three cases: ( a ) x = 1 ( b ) x = q ( c ) x / ∈ ± q Z . Proof. We sketch a proof, in the same spirit as in the introduction of [20]. We deduce fromTheorem 6.19 that it suffices to study the cyclotomic quotients of b H ( D n ) given by a relation Y i ∈ I x ( X − i ) Λ i , here I x and Λ are as in the proof of Corollary 6.9. By Theorem 6.17, this cyclotomicquotient is only determined by the quiver Γ and its involution θ as defined in the proof ofCorollary 6.9. In particular, looking at the number of connected components of Γ we stillhave the two cases x ∈ ± q Z (which give cases ( a ) and ( b ) ) and x / ∈ ± q Z (which is case ( c ) ). In the latter case all the choices of x lead to isomorphic algebras since θ has no fixedpoints. Remark . We make an additional final remark on the three cases ( a ) – ( c ) to be considered,similarly to Remark 6.10. Cases ( a ) and ( b ) correspond to a quiver with a single connectedcomponent (an infinite oriented line or a finite oriented polygon depending on whether q is a root of unity or not), while ( c ) corresponds to a quiver with two identical connectedcomponents exchanged by the involution θ . Case ( a ) corresponds to θ having a fixed point,while Case ( b ) generically corresponds to the situation where there is no fixed point. Asbefore, when q is an odd root of unity, Case ( b ) is not necessary since it is equivalent toCase ( a ) . A Polynomial realisation We prove here Lemma 3.25. In this appendix, for any f ∈ K [ x, β ] we also systematicallywrite f for the the element of End K ( K [ x, β ]) given by left multiplication and we use con-catenation to denote the composition inside End K ( K [ x, β ]) . In particular, for any w ∈ B n and f ∈ K [ x ] we have wf = ( w f ) w inside End K ( K [ x, β ]) .We now define some elements of End K ( K [ x, β ]) by ϕ ( e ( i )) = i ,ϕ ( y a e ( i )) = x a i ,ϕ ( ψ b e ( i )) = (cid:16) δ i b ,i b +1 ( x b − x b +1 ) − ( r b − 1) + P i b ,i b +1 ( x b +1 , x b ) r b (cid:17) i ,ϕ ( ψ e ( i )) = (cid:0) γ i x − (1 − r ) + α i ( x ) r (cid:1) i , (A.1)for any a ∈ { , . . . , n } and b ∈ { , . . . , n − } , and extend these formulas to ϕ ( X ) for X ∈ { y , . . . , y n , ψ , . . . , ψ n − } by ϕ ( X ) = P i ∈ β ϕ ( Xe ( i )) .We will prove that ϕ extends to an algebra homomorphism ϕ : V β (Γ , λ, γ ) → End K ( K [ x, β ]) ,which will imply Lemma 3.25. Indeed, the map ϕ is the homomorphism associated with theaction defined in §3.2. To prove that ϕ extends to an algebra homomorphism, we check thedefining relations of V β (Γ , λ, γ ) . Recall that P i,j = 0 when i = j so that ϕ ( ψ b e ( i )) = ( ( x b − x b +1 ) − ( r b − i , if i b = i b +1 ,P i b ,i b +1 ( x b +1 , x b ) r b i , otherwise . Moreover, by (3.5a) and (3.24) we have ϕ ( ψ e ( i )) = ( α i ( x ) r i , if γ i = 0 ,γ i x − (1 − r ) i , otherwise . The relations that do not involve ψ are satisfied since the action is the same as in [23,Proposition 3.12]. Relations (3.9), (3.10) and (3.12) are immediate.To simplify the notation, for any v ∈ V β (Γ , λ, γ ) we also write v ′ instead of ϕ ( v ) . Notethat the composition operation in End K ( K [ x, β ]) is denoted as a simple multiplication. Forexample, ψ ′ x means composition of the multiplication by x with the operator φ ( ψ ) .Concerning (3.11), we have ( ψ ′ y ′ + y ′ ψ ′ ) e ( i ) ′ = ψ ′ x i + x (cid:0) γ i x − (1 − r ) + α i ( x ) r (cid:1) i = h(cid:0) γ i x − (1 − r ) x + α i ( x ) r x (cid:1) + ( γ i (1 − r ) + x α i ( x ) r ) i i = h γ i (1 + r ) − x α i ( x ) r + γ i (1 − r ) + x α i ( x ) r i i = 2 γ i i = ϕ (cid:0) γ i e ( i ) (cid:1) . or (3.13), if γ i = 0 then γ θ ( i ) = 0 by (3.6) and we have, noting that j r = r r · j inside End K ( K [ x, β ]) , ψ ′ e ( i ) ′ = ψ ′ α i ( x ) r i = α θ ( i ) ( x ) r α i ( x ) r i = α θ ( i ) ( x ) α i ( − x ) i = ( − λ θ ( i x d ( i )1 i = ϕ (cid:16) ( − λ θ ( i y d ( i )1 e ( i ) (cid:17) , by (3.23), and if γ i = 0 then γ θ ( i ) = 0 and we have ψ ′ e ( i ) ′ = ψ ′ γ i x − (1 − r ) i = γ θ ( i ) γ i (cid:0) x − (1 − r ) (cid:1) i = 0 . It remains to check (3.14). As in (3.20), we write i i and even instead of i , and ¯ a insteadof θ ( i a ) . We have, using (3.9), ( ψ ′ ψ ′ ) e (12) ′ = ( ψ ′ )( ψ ′ ¯21 )( ψ ′ )( ψ ′ ) , (A.2a) ( ψ ′ ψ ′ ) e (12) ′ = ( ψ ′ ¯2¯1 )( ψ ′ )( ψ ′ ¯12 )( ψ ′ ) . (A.2b) A.1 Case γ i = 0 = γ i First, recall that by (3.6) we know that if γ i = 0 and θ ( i ) = i then γ i = 0 . Thus, wewant to prove that (cid:0) ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) (cid:1) i = ( ( − λ θ ( i ( − y ′ ) d ( i − y ′ d ( i y ′ + y ′ ψ ′ i , if θ ( i ) = i , , otherwise . (A.3)Since γ i = γ θ ( i ) = γ i = γ θ ( i ) = 0 , for any a, b ∈ { , , ¯1 , ¯2 } the element ψ acts on ab as α a ( x ) r .Assume that θ ( i ) = i and θ ( i ) = i . By (3.5b) we have d ( i ) = d ( i ) = 0 , thus (A.3)becomes (cid:0) ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) (cid:1) i = 0 . (A.4)Since d ( i ) = d ( i ) = 0 , by (3.23) we can assume α i ( y ) = α i ( y ) = 1 , thus ψ acts on ab as r for any a, b . Hence, the same calculation as in [20, §3.1] proves that (A.4) is satisfied.In the opposite case, if θ ( i ) = i and θ ( i ) = i we know by the proof of [25, Proposition7.4] that (A.3) holds.Thus, we now assume that θ ( i ) = i and θ ( i ) = i , in particular i = i and θ ( i ) = i .As above, we have d ( i ) = 0 thus ψ acts on a as r . We obtain from (A.2), omitting theidempotents, ( ψ ′ ψ ′ ) = r P ¯21 ( x , x ) r α ( x ) r P ( x , x ) r = P ¯21 ( x , − x ) r α ( x ) r r P ( x , x ) r = P ¯21 ( x , − x ) α ( x ) P ( − x , − x ) r r r r , and ( ψ ′ ψ ′ ) = P ¯2¯1 ( x , x ) r α ( x ) r P ¯12 ( x , x ) r r = P ¯2¯1 ( x , x ) α ( x ) r r P ¯12 ( x , x ) r r = P ¯2¯1 ( x , x ) α ( x ) P ¯12 ( x , − x ) r r r r , thus ( ψ ′ ψ ′ ) = ( ψ ′ ψ ′ ) as desired, where we used ¯1 = 1 and (3.21). The case θ ( i ) = i and θ ( i ) = i is similar. Remark A.5 . (See Remark 3.7.) Without condition (3.5b), we have to choose another, morecomplicated, relation (3.14), if we want it to be compatible with the action on polynomials. .2 Case γ i = 0 = γ i We want to prove that (cid:0) ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) (cid:1) i = γ i Q i i ( y ′ , − y ′ ) − Q i i ( y ′ , y ′ ) y ′ ψ ′ i , that is, (cid:0) ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) (cid:1) i = γ i Q i i ( x , − x ) − Q i i ( x , x ) x α i ( x ) r i . By (3.5a) we have θ ( i ) = i . Note that γ i = 0 = γ i implies i = i . By (A.2) we have,omitting the idempotents, ( ψ ′ ψ ′ ) = α ( x ) r P ¯21 ( x , x ) r γ x − (1 − r ) P ( x , x ) r = α ( x ) P ¯21 ( x , − x ) γ x − r r (1 − r ) P ( x , x ) r = α ( x ) P ¯21 ( x , − x ) γ x − (cid:2) P ( − x , x ) r r − P ( − x , − x ) r r r (cid:3) r = γ x − α ( x ) P ¯21 ( x , − x ) (cid:2) P ( − x , x ) r r − P ( − x , − x ) r r r (cid:3) r , and ( ψ ′ ψ ′ ) = P ¯2¯1 ( x , x ) r γ x − (1 − r ) P ¯12 ( x , x ) r α ( x ) r = P ¯2¯1 ( x , x ) γ x − r (1 − r ) P ¯12 ( x , x ) α ( x ) r r = P ¯2¯1 ( x , x ) γ x − (cid:2) P ¯12 ( x , x ) r − P ¯12 ( x , − x ) r r (cid:3) α ( x ) r r = P ¯2¯1 ( x , x ) γ x − α ( x ) (cid:2) P ¯12 ( x , x ) r − P ¯12 ( x , − x ) r r (cid:3) r r = γ x − α ( x ) P ¯2¯1 ( x , x ) (cid:2) P ¯12 ( x , x ) − P ¯12 ( x , − x ) r r r (cid:3) r . Thus, recalling ¯2 = 2 and using the properties (2.2), (3.3), (3.21), (3.22) for the families P and Q we have ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) = γ x − α ( x ) (cid:2) P ¯21 ( x , − x ) P ( − x , x ) − P ¯2¯1 ( x , x ) P ¯12 ( x , x ) (cid:3) r = γ x − (cid:2) Q ( x , − x ) − Q ( x , x ) (cid:3) α ( x ) r = γ x − (cid:2) Q ( x , − x ) − Q ( x , x ) (cid:3) α ( x ) r , as desired. A.3 Case γ i = 0 = γ i We want to prove that (cid:0) ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) (cid:1) i = 0 . Similarly to §A.2 we have θ ( i ) = i = i . By (A.2) we have, omitting the idempotents, ( ψ ′ ψ ′ ) = γ x − (1 − r ) P ¯21 ( x , x ) r α ( x ) r P ( x , x ) r = γ x − [ P ¯21 ( x , x ) − P ¯21 ( x , − x ) r ] α ( x ) P ( x , − x ) r r r = γ x − α ( x ) [ P ¯21 ( x , x ) P ( x , − x ) − P ¯21 ( x , − x ) P ( − x , − x ) r ] r r r = γ x − α ( x ) P ¯21 ( x , x ) P ( x , − x )(1 − r ) r r r by (3.21), and ( ψ ′ ψ ′ ) = P ¯21 ( x , x ) r α ( x ) r P ( x , x ) r γ x − (1 − r )= γ x − α ( x ) P ¯21 ( x , x ) P ( x , − x ) r r r (1 − r ) , Thus ( ψ ′ ψ ′ ) = ( ψ ′ ψ ′ ) as desired. .4 Case γ i = 0 = γ i We want to prove that (recalling from (2.1) that Q ii = 0 ) (cid:0) ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) (cid:1) i = ( γ i Q i i ( y ′ , − y ′ ) − Q i i ( y ′ ,y ′ ) y ′ y ′ ( y ′ ψ ′ − γ i ) i , if i = i , , otherwise , that is, since ψ acts on i as γ i x − (1 − r ) (recalling that θ ( i ) = i by (3.5a)), (cid:0) ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) (cid:1) i = ( γ i γ i Q i i ( x ,x ) − Q i i ( x , − x ) x x r i , if i = i , , otherwise . The next result is an easy calculation. Lemma A.6. Let P be a polynomial in x , x and let w ∈ h r , r i . Then x − (1 − r ) P w − P wx − (1 − r ) = (cid:0) x − − w x − (cid:1) P w + w x − P wr − x − r P r w , inside End K ( K [ x, β ]) . By (3.5a) we have θ ( i ) = i . If i = i we obtain from (A.2) ψ ′ ψ ′ ψ ′ = P ( x , x ) r γ x − (1 − r ) P ( x , x ) r = γ x − P ( x , x ) r (1 − r ) P ( x , x ) r = γ x − P ( x , x ) r (cid:2) P ( x , x ) − P ( x , − x ) r (cid:3) r = γ x − P ( x , x ) (cid:2) P ( x , x ) r − P ( x , − x ) r r (cid:3) r = γ x − P ( x , x ) (cid:2) P ( x , x ) − P ( x , − x ) r r r (cid:3) . Since ψ ′ = γ x − (1 − r ) , we can apply Lemma A.6 for the two above summands. Weobtain that second summand will vanish in (cid:0) ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) (cid:1) since x − ∈ K ( x ) is in-variant under r r r and P ( x , x ) P ( x , − x ) ∈ K [ x , x ] is invariant under r by (3.21).Thus, we only consider the first summand, which is equal to γ x − Q ( x , x ) , and weobtain, omitting the idempotents and using (2.2) and (3.3), ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) = γ γ x − x − (cid:2) Q ( x , x ) − Q ( x , − x ) (cid:3) r = γ γ x − x − (cid:2) Q ( x , x ) − Q ( x , − x ) (cid:3) r , as desired.Finally, assume that i = i . We have ( ψ ′ ψ ′ ) − ( ψ ′ ψ ′ ) = γ (cid:2) x − (1 − r )( x − x ) − ( r − x − (1 − r )( x − x ) − − ( x − x ) − ( r − x − (1 − r )( x − x ) − x − (1 − r ) (cid:3) = 0 , since this is just the braid relation for the divided difference operators ∂ := x − (1 − r ) and ∂ := ( x − x ) − ( r − (see [3, 7]). References [1] S. Ariki , On the decomposition numbers of the Hecke algebra of G ( m, , n ) . J. Math.Kyoto Univ. (1996) 789–808.[2] S. Ariki and K. 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