the Double Affine Hecke Algebras

45(2009), 845–905

On Hecke Algebras Associated with Elliptic Root Systems and

the Double Affine Hecke Algebras

Dedicated to Masaki Kashiwara on the occasion of his sixtieth birthday

By

Yoshihisa Saito∗and MidoriShiota∗

Abstract

We define the elliptic Hecke algebras for arbitrary marked elliptic root systems in terms of the corresponding elliptic Dynkin diagrams and make a ‘dictionary’ between the elliptic Hecke algebras and the double affine Hecke algebras.

§1. Introduction

1.1. Over the last fifteen years or so, there were remarkable developments in the study of multi-variable orthogonal polynomials, attached to root systems.

One of these developments was due to Cherednik. In [C1], he defined an dif- ference analogue of Knizhnik-Zamolodikov equations, so-called affine quantum difference Knizhnik-Zamolodikov equations and established their equivalence with the corresponding eigenvalue problem of Macdonald type. To prove the above equivalence, he introduced a new class of algebras, so-called thedouble affine Hecke algebras. Moreover, he proved Macdonald’s inner product conjec- ture in [C2]. In a process of solving it, the double affine Hecke algebras also played an important role.

Cherednik’s construction is generalized to an important class of non- reduced root systems, (C_n^∨, C_n) by Noumi [N] and Sahi [Sa]. When n = 1

Communicated by M. Kashiwara. Received November 14, 2007. Revised September 1, 2008, December 12, 2008, January 30, 2009.

2000 Mathematics Subject Classification(s): Primary 17B35; Secondary 14D30, 16G20.

∗Graduate School of Mathematical Sciences, University of Tokyo, Tokyo 153-8914, Japan.

e-mail: yosihisa@ms.u-tokyo.ac.jp

c 2009 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.

(rank 1 case), the corresponding orthogonal polynomials are the Askey-Wilson polynomial [AW] which include as special and limiting cases all the classical families of orthogonal polynomials in one variable. In [M5], Macdonald formu- lated all the above results uniformly.

1.2. In the middle of 1980’s, K. Saito [S] defined a notion of the marked elliptic root systems which is a generalization of finite or affine root systems, motivated by the study of simple elliptic singularities. Attaching each marked elliptic root system, he introduced a diagram, so-called the elliptic Dynkin diagram which describes the structure of a marked elliptic root system. In addition, he gave a complete classification of marked elliptic root systems under some suitable assumptions. In the original motivation, vertices in an elliptic Dynkin diagram correspond to vanishing cycles and edges describe intersection numbers of them.

After K. Saito’s work, he and Takebayashi studied the structure of the Weyl groups associated to marked elliptic root systems, so-called the elliptic Weyl groups[ST]. In particular, they found a new presentation of elliptic Weyl groups in terms of the corresponding elliptic Dynkin diagrams. The explicit meaning is as follows. In the finite and affine cases, it is well-known that the structure of the Weyl groups can be described by the corresponding Coxeter-Dynkin diagrams. Namely, the set of generators and relations of the Weyl group can be read from the corresponding Coxeter-Dynkin diagram. As a generalization, they gave a generating system of the elliptic Weyl group attached to vertices of the elliptic Dynkin diagram and the defining relation which are described by the ‘shape’ of it. These relations are called the elliptic Coxeter relations.

Since the Weyl groups of finite and affine root systems are Coxeter groups, one can consider the corresponding Hecke algebras. In the elliptic case, as an application of the K. Saito-Takebayashi’s presentation, Yamada [Y] defined a q-analogue of elliptic Weyl groups called the elliptic Hecke algebras for “one- codimensional” marked elliptic root systems which have only one dotted line in their elliptic Dynkin diagrams. After that Takebayashi [T1], [T2] defined them for arbitrary marked elliptic root systems except for the group(D)(c.f.

4.2). Yamada and Takebayashi also pointed out that elliptic Hecke algebras are much like double affine Hecke algebras. More precisely, for some cases, they stated that the elliptic Hecke algebras are embedded into the double affine Hecke algebras.

1.3. The aim of this article to establish an explicit connection between

the elliptic Hecke algebras and the double affine Hecke algebras. For that purpose, we reformulate the uniform construction of the double affine Hecke algebras due to Macdonald [M5]. In Section 2, we give a quick review of the theory of the affine Hecke algebras. All statements in this section are well- known. In Section 3, we introduce a notion oftriplets. This is a basic datum to define the double affine Hecke algebras and a key of our construction. For a giving triplet, we define the double affine Hecke algebras and give some basic properties of them. After recalling the theory of elliptic root systems in Section 4 following K. Saito [S], we give the definition of the elliptic Hecke algebras in Section 5. They are defined by some generators and relations attached to the elliptic Dynkin diagramsof the corresponding (marked) elliptic root system. In addition, we give another presentation of them. (Proofs of the statements are given in Section 7.) Section 6 is the main part of this article. For a giving marked elliptic root system (R, G), we introduce the corresponding triplet and the double affine Hecke algebra attached to it as in Section 3. On the other hand we have another algebra (the corresponding elliptic Hecke algebra) attached to (R, G) as in Section 5. After that, we make a comparison between them. This is a main result of this article (Theorems 6.2.3, 6.3.2).

1.4. Finally, we must refer the results of Takebayashi. As we already mentioned above, he introduced a notion of the elliptic Hecke algebras. More precisely, in [T1], he defined them for elliptic root systems of type (1,1) and compare them and the double affine Hecke algebras by case-by-case checking.

After that, in [T2], he defined them for arbitrary marked elliptic root systems except for the group (D) (c.f. 4.2), but he did not compare them and the double affine Hecke algebras for arbitrary cases. In his definition, he use new diagrams which are called the “completed elliptic Dynkin diagrams”. But, as we mentioned above, the elliptic Dynkin diagram have a concrete meaning in a geometrical setting. Therefore, in this article, we try to ‘re-define’ ellip- tic Hecke algebras by using the original elliptic Dynkin diagrams, in stead of the completed elliptic Dynkin diagrams and to make an explicit and uniform

‘dictionary’ between the elliptic Hecke algebras and the double affine Hecke algebras for arbitrary cases.

The announcement of the results of this article already appeared as [SS].

§2. Affine Hecke Algebras

2.1. Affine root systems and affine Weyl groups. Let V be an n-

dimensional real vector space with a positive definite symmetric bilinear form

·,·, R₀ ⊂V an irreducible finite root system and fixa₁,· · ·, a_n simple roots in R₀. For a ∈ R₀ set a^∨ := 2a/a, a. Denote by Q(R₀) = ⊕Za_i the root lattice, (Q(R₀))₊ =⊕Z≥0a_i, P(R₀) = {λ ∈V | λ, a^∨_i ∈Z} the weight lat- tice, (P(R₀))₊ the set of all dominant weights, (P(R₀))₋ =−(P(R₀))₊ and W(R₀) the corresponding Weyl group. SetR₀^∨={a^∨ | a∈R₀}. It is also an irreducible finite root system.

LetF:=V⊕Rcand we will interpret an element ofFas a function onV by (u+rc)(v) =u, v+r. We extend·,·to a positive semidefinite bilinear form onFbyu₁+r₁c, u₂+r₂c:=u₁, u₂. LetS(R₀) be the set of all vectors of the forma+rcwherea∈R₀andris any integer if ¹₂a∈R₀(resp. any odd integer if ¹₂a∈R₀). Set a₀:=−θ+c , whereθ is the highest root ofR₀. ThenS(R₀) is an irreducible reduced affine root system with simple roots a₀, a₁,· · ·, a_n. We remark thatccan be written in the following form: c=_n

i=0n_ia_i, where n_i ∈Z>0 and n₀ = 1. The dual root system S(R₀)^∨ :={a^∨ | a∈ S(R₀)} is also an irreducible reduced affine root system with a basisa^∨₀,· · ·, a^∨_n.

For later use, we introduce the following notation: set b_i :=

a_i, ifS =S(R₀), a^∨_i, ifS =S(R₀)^∨.

ForS =S(R₀) orS(R₀)^∨, we denote byQ(S) :=⊕ⁿ_i=0Zb_iits root lattice.

IfR₀ of typeX whereX is one of the symbolsA_n,B_n,C_n,BC_n,D_n,E₆, E₇,E₈,F₄,G₂, we say thatS(R₀) (resp. S(R₀)^∨) is of typeX(resp. X^∨). It is known that any irreducible reduced affine root systemS is isomorphic to either S(R₀) or S(R₀)^∨. In Appendix, we will present a complete list of irreducible reduced affine root systems.

Firstly assume thatS is an irreducible reduced affine root system. Namely S=S(R₀) orS(R₀)^∨. LetW(S) be the affine Weyl group ofS. It is generated by reflectionsw_f (f ∈S) wherew_f(g) =g− g, f^∨f forg∈F. Since (f^∨)^∨= f, we havew_f∨ =w_f andW(S(R₀)) =W(S(R₀)^∨). Define the action ofv∈V inFbyt(v) :f →f− f, vc. The following fact is well-known.

Theorem 2.1.1. (1) W(S) =W(R₀)t(Q(R^∨₀)). (2)W(S)is gener- ated byw_i:=w_bi (i= 0,· · ·, n)and a Coxeter group which corresponds to the affine Dynkin diagram ofS.

Let ˜W(S) := W₀t(P(R^∨₀)) be the extended affine Weyl group. It is easy to see that W(S) is a normal subgroup of ˜W(S) and ˜W(S)/W(S) ∼=

P(R^∨₀)/Q(R^∨₀). LetS⁺ be the set of positive roots andS⁻ :=−S⁺. Forw∈ W˜(S), definel(w) :=|S⁺∩w⁻¹S⁻|. Ifw∈W(S), its length with respect to the generatorsw₀,· · · , w_nis just equal tol(w). Define Ω :={w∈W˜(S)|l(w) = 0}. It is a subgroup and ˜W(S) = ΩW(S). Therefore Ω∼=P(R^∨₀)/Q(R^∨₀). Since Ω is a subgroup of ˜W(S), it acts onS. Moreover, it is known that, Ω preserves the set of all simple roots. Therefore, foru∈Ω such thatu(b_i) =b_j, we have uw_iu⁻¹=w_j.

For later use, we will explain an explicit structure of Ω. Let v_i be the shortest element of W(R₀) such that v_iω_i ∈ P(R^∨₀)₋, where {ω_i}ⁿ_i=1 is the set of all fundamental weights ofP(R^∨₀). Letu_i =t(ω_i)v_i⁻¹ (1 ≤i≤n) and u₀= 1.

Lemma 2.1.2. Set J = {j | 0 ≤ j ≤ n, n_j = 1}. We have Ω = {u_j | j∈J}.

Remark. We have already defined W(S) and ˜W(S) for any irreducible reduced affine root system (not only forS =S(R₀)). As we mentioned above, W(S(R₀)) =W(S(R₀)^∨). Moreover, by the construction, we have ˜W(S(R₀)) = W˜(S(R₀)^∨).

Secondly assume S is an irreducible, non-reduced affine root system. In this case, the following fact is known:

Fact 1. Let S₁:={a∈S |a/2 ∈S} and S₂:={a∈S | 2a∈S}. We haveS=S₁∪S₂ and bothS₁andS₂ are reduced affine root systems with the same affine Weyl group.

We say thatS is of type (X₁, X₂) whereX_iis the type ofS_i (i= 1,2). In this case, the basis of S is that of S₁ and its affine Weyl group W(S) is equal to W(S₁) =W(S₂).

2.2. Affine Hecke algebras. In this subsection, we assumeS is reduced.

Definition 2.2.1. (1) Let ˜Bthe group with generatorsT(w) (w∈W˜(S)) and relations:

T(v)T(w) =T(vw), ifl(v) +l(w) =l(vw).

(2) LetB be the subgroup of ˜Bgenerated byT_i:=T(w_i) (i= 0,· · ·, n).

We writeU_j =T(u_j) forj∈J. It is known that ˜Bis generated byT_i,U_j.

Consider a Laurent polynomial ringZ[τ₀^±1,· · ·, τ_n^±1]. Let ˜I (resp. I) be the ideal generated by the elementsτ_i−τ_j where w_i andw_j are conjugate in W˜(S) (resp. W(S)). Set

A˜_a=Z[τ₀^±1,· · ·, τ_n^±1]/I˜ and A_a=Z[τ₀^±1,· · · , τ_n^±1]/I.

Obviously both ˜A_a and A_a are isomorphic to some Laurent polynomial rings in several variables. More precisely, if R₀ is simply laced, ˜Aa is a Laurent polynomial ring in one variable. If not, ˜Aa have two variables; one corresponds to short roots and the other to long roots. IfR₀is not of typeA₁orC_n, ˜I =I. Therefore we have ˜A_a = A_a. In the case of type A₁, A_a has two variables.

These are τ₀, τ₁ which correspond to simple roots a₀ and a₁. In the case of typeC_n,A_a has three variables. These areτ₀,τ_n andτ₁=· · ·=τ_n−1. Here a₀anda_n are long simple roots and the others are short simple roots.

Definition 2.2.2. (1) The extended affine Hecke algebra H( ˜W(S)) is the quotient of the group algebra ˜A_a[B] by the ideal generated by the following relations:

(A1) (T_i−τ_i)(T_i+τ_i⁻¹) = 0, fori= 0,· · · , n.

(2) The affine Hecke algebra H(W(S)) is the quotient of the group algebra A_a[B] by the ideal generated by the same relations as (A1).

We regard ˜Aaas anAa-algebra via a natural projectionAa→A˜a. The ˜Aa- algebra ˜Aa⊗AaH(W(S)) is naturally isomorphic to the subalgebra ofH( ˜W(S)) which is generated byT_i (i= 0,· · ·, n).

Theorem 2.2.3. Under the convention which we mentioned above, we haveH( ˜W(S)) = Ω( ˜A_a⊗_AaH(W(S))), where the action ofΩ onT_i is the same as Weyl group case.

There is another presentation of H( ˜W(S)) which is very useful to study affine Hecke algebras. Forμ∈P(R^∨₀) defineY^μ ∈ H( ˜W(S)) as follows: (i) If μ ∈P(R^∨₀)₊, thenY^μ :=T(t(μ)); (ii) Ifμ =λ−ν withλ, ν ∈P(R^∨₀)₊, thenY^μ :=T(t(λ))T(t(ν))⁻¹.

For H(W(S)), we also define Y^μ ∈ H(W(S)) in the similar way by re- placingP(R^∨₀) withQ(R^∨₀).

We introduce the following notation: let

b(z₁, z₂;x) =z₁−z₁⁻¹+ (z₂−z₂⁻¹)x

1−x² ,

where z₁, z₂ and x are indeterminates. When z₁ = z₂, b(z₁, z₂;x) has the simpler form

b(z₁, z₁;x) =z₁−z₁⁻¹ 1−x .

In the case ofH( ˜W(S)) such thatR₀is of typeA₁or of typeC_n, for 1≤i≤n, we set

τ_i=

τ_i, i=n, τ₀, i=n.

For the other case, we setτ_i=τ_i for anyi= 1,· · · , n.

Theorem 2.2.4. (1) Y^μ is well-defined for allμ and

(A2) Y^μY^ν =Y^μ+ν.

(2) In the algebra H( ˜W(S)) (resp. H(W(S))), the following relations hold (called Lusztig’s relations) :

(A3) Y^μT_i−T_iY^wi(μ)=b(τ_i, τ_i;Y^−a∨i)(Y^μ −Y^wi(μ)), fori= 1,· · ·, nandμ∈P(R₀^∨) (resp. Q(R^∨₀)).

(3) Let us consider the algebra generated by T_i (i = 1,· · · , n) and Y^μ (μ ∈ P(R^∨₀))and relations(A1)fori= 1,· · · , n,(A2)and(A3). Then it is isomor- phic toH( ˜W(S)). Further, by replacingP(R^∨₀)withQ(R^∨₀), the corresponding algebra is isomorphic toH(W(S)).

§3. Double Affine Hecke Algebras

In this section, we give the definition of the double affine Hecke algebras in terms of triplets. We remark that our definition is not new. It is only a reformulation of the uniform construction of the double affine Hecke algebras due to Macdonald [M5].

3.1. Triplets. Let us consider the following three types of datum Ξ = (R₀;S, Λ_s) which we call a triplet:

(type I) R₀ is a finite irreducible reduced root system, S=S(R₀) orS(R₀)^∨, Λ_s=Q(S) whereS =S(R₀).

(type II)

R₀ andS are as the same in type I, Λ_s=Q(S) whereS=S(R₀)^∨.

(type III) R₀is of typeC_n (n≥1) (Here we denoteC₁=A₁.);

S is of type (C_n^∨, C_n), Λ_s=Q(S(R₀)^∨).

(In Appendix, we present the detailed structure of the affine root system of type (C_n^∨, C_n).)

Set L=

⎧⎪

⎨

⎪⎩

P(R₀), (type I), P(R^∨₀), (type II), Q(R^∨₀), (type III),

L=

⎧⎪

⎨

⎪⎩

P(R^∨₀), (type I), P(R^∨₀), (type II), Q(R^∨₀), (type III).

In each case, let Λ :=L⊕Zc₀. Herec₀ =e⁻¹c and e is the exponent of Ω∼=P(R^∨₀)/Q(R₀^∨),

Next we fix a normalization of·,·. Recall a basis{a_i}ⁿ_i=0 ofS(R₀). For type I and II, we normalize·,·asθ, θ= 2. Therefore we havea^∨₀ =−θ+c= a₀. For type III, S =S₁∪S₂ whereS₁=S(R₀)^∨ and S₂ =S(R₀). Here R₀ is of typeC_n (n≥1). In this casee= 2 and we normalize ·,·as θ, θ= 4.

Therefore we have a₀, a₀=a_n, a_n= 4 anda_i, a_i= 2 (i= 1,· · · , n−1).

Moreover a basis{a^∨_i}ofS₁and a basis {a_i}ofS₂are related by the following way:

a^∨₀ =1

2a₀=−θ 2+c

2, a^∨_n =1

2a_n, a^∨_i =a_i (i= 1,· · ·, n−1).

Under the above convention, we immediately have the following lemma.

Lemma 3.1.1. For any case, Λ_s is a sublattice ofΛ.

Let us introduce the following notations:

a_i:=

a_i, (type I), a^∨_i, (type II or III).

We remark that{a_i}ⁿ_i=0is a basis of Λ_s. In each case, let

W(Ξ) :=W(R₀)t(L), and W(Ξ)_s:=W(R₀)t(Q(R^∨₀)).

If Ξ is of type I or II, the first one is the extended affine Weyl group ofS and the second is non-extended one. On the other hand, for type III, both are the

affine Weyl group ofS.

The first statement of the following lemma is due to Macdonald [M5] and the second is trivial by the definition.

Lemma 3.1.2. (1) Λis stable under the action of W(Ξ).

(2) Λ_s is stable under the action ofW(Ξ)_s.

3.2. Definition of double affine Hecke algebras. LetAbe a commutative ring defined by the following way:

A=

A˜_a, (type I or II), Aa[(τ₀)^±1,(τ_n)^±1], (type III), whereτ₀ andτ_n are new indeterminates.

Definition 3.2.1. Let Ξ = (R₀;S,Λ_s) be a triplet given in the previous subsection. Fori= 0,· · · , n, letb_i(x) =b(τ_i, τ_i;x).Here we setτ_i=τ_ifor all iwhen Ξ is of type I or II and fori= 0, nwhen Ξ is of type III.

The double affine Hecke algebra H(Ξ) is an associative A-algebra defined by the following way.

If Ξ is of type I or II, it is generated by T_i (i = 0,· · · , n), U_j (j ∈ J), X^λ (λ∈Λ) subject to the following relations:

(D1) T_i andU_j satisfy the same relations inH(W(S)),

(D2) X^λX^μ=X^μX^λ=X^λ+μ,

(D3) T_iX^λ−X^wi(λ)T_i=b_i(X^ai)(X^λ−X^wi(λ)),

(D4) U_jX^λU_j⁻¹=X^uj(λ).

If Ξ is of type III, it is generated by T_i (i= 0,· · ·, n),X^λ(λ∈Λ) subject to the similar relations as (D1), (D2), (D3).

Following [M5], we say H(Ξ) is the double affine Hecke algebra of type (S, S) for a triplet Ξ of type I or II. For a triplet of type III, we sayH(Ξ) is the double affine Hecke algebra of type (C_n^∨, C_n).

The following theorem is essentially due to Macdonald [M5].

Theorem 3.2.2. (1)X^c0 is a central element. (2)Each of the following sets

{X^λT(w)Y^μ |λ∈Λ, w∈W(R₀), μ∈L}, {Y^μT(w)X^λ |λ∈Λ, w∈W(R₀), μ∈L},

{X^λT(w)| λ∈Λ, w∈W(Ξ)}, {T(w)X^λ |λ∈Λ, w∈W(Ξ)}

forms a freeA-basis of H(Ξ).

Remark. In the original article [M5], the definition ofH(Ξ) is slightly different: eis the positive integer such thatL, L=e⁻¹Z, except in type III in which case e = 2. The central element q₀ := X^c0 is considered as a real number such that 0 < q₀ < 1. τ_i and τ_i are also considered as positive real numbers. Assumeq₀,τ_iandτ_iare algebraically free overZandH(Ξ) is defined as an algebra overKwith same generators and relations, whereK is a subfield ofR containingq₀, all τ_i and τ_i. The original theorem is the following: each of the four sets which is given by replacing Λ withL in the above theorem is aK-basis ofH(Ξ). But, in our situation, we can prove our statements by the similar argument. So we omit the proof.

Definition 3.2.3. The small double affine Hecke algebraH(Ξ)_sis the subalgebra ofH(Ξ) which is generated byT(w) (w∈W(Ξ)_s) andX^λ(λ∈Λ_s).

We remark thatH(Ξ)_s is just equal toH(Ξ) for Ξ of type III.

Assume Ξ of type I or II. By Lemma 3.1.1 (2), Definition 3.2.1 and Theorem 3.2.2, we immediately have the following statement.

Corollary 3.2.4. (1)The similar relations as (D1), (D2)and (D3)in Definition 3.2.1hold in H(Ξ)_s. (2)Each of the following sets

{X^λT(w)Y^μ |λ∈Λ_s, w∈W(R₀), μ ∈Q(R₀)^∨}, {Y^μT(w)X^λ |λ∈Λ_s, w∈W(R₀), μ ∈Q(R₀)^∨},

{X^λT(w)| λ∈Λ_s, w∈W(Ξ)_s}, {T(w)X^λ |λ∈Λ_s, w∈W(Ξ)_s} forms a freeA-basis of H(Ξ)s.

Therefore we have the following.

Corollary 3.2.5. As a set H(Ξ)/H(Ξ)_s ∼= (Λ/Λ_s) ×Ω. Especially H(Ξ)_s is a subalgebra ofH(Ξ)with a finite index.

§4. Summary of Elliptic Root Systems

4.1. Marked Elliptic root systems. LetF be an (n+ 2) dimensional real vector space with a positive semi-definite symmetric bilinear formI:F×F →R with the two-dimensional radical which is denoted by rad(I). Ifα∈F satisfies I(α, α)= 0, we sayαis a non isotropic vector. For a non isotropic vectorα∈F, putα^∨:= 2α/I(α, α) and define a reflections_α bys_α(u) :=u−I(u, α^∨)αfor u∈F.

Definition 4.1.1. A set R of non isotropic vectors in F is called an elliptic root system of rank n if the following conditions are satisfied: (i) Q(R)⊗ZR ∼= F. (Here Q(R) is the additive subgroup of F generated by R.) (ii)s_α(R) =Rfor anyα∈R. (iii)I(α, β^∨)∈Zfor anyα, β∈R. (iv)Ris irreducible. That is, there exists no partition ofRinto two non-empty subsets R₁ andR₂ such thatI(α, β) = 0 for allα∈R₁ andβ∈R₂.

Let W(R) be the group generated by all reflections s_α (α∈R). We call W(R) the elliptic Weyl group.

A subspace Gof rad(I) of rank 1 defined over Qis called a marking and the pair (R, G) is called a marked elliptic root system.

We fix a generatorδ₁of the rank 1 latticeG∩Q(R): G∩Q(R) =Zδ₁.For α∈R, setk_α:= inf{k∈Z>0 |α+kδ₁∈R} andα^∗:=α+k_αδ₁.

Let π_a :F → F/G(resp. π_f :F →F/rad(I)) be the natural projection and set R_a := π_a(R) (resp. π_f(R) := R_f). R_a (resp. R₀) is an affine (resp.

finite) root system. In the present paper we assume thatR_f is reduced, which implies thatR_a is also reduced.

We fix a subset Γ_a={α₀,· · · , α_n}ofR such thatπ_a(Γ_a) forms a basis of the affine root systemR_a. Letδ_a be the primitive imaginary root ofR_a. Then δ_a can be written in the following form: δ_a =_n

i=0n_iπ_a(α_i), (n_i ∈ Z>0).It is well-known that there always exists an elementα ∈ Γ_a, say α₀, such that n_α0 = 1. Setδ₂:=_n

i=0n_iα_i∈R andθ:=_n

i=1n_iα_i.

By the construction it is easy to see thatQ(R) has a following expression:

Q(R) = n

i=0⊕ Zα_i ⊕Zδ₁=n

i=1⊕ Zα_i ⊕Zδ₁⊕Zδ₂.

For 0≤i≤n, setm_i:=I_R(α_i, α_i)n_i/2k_αi, whereI_Ris a constant multiple ofI normalized such that inf {I_R(α, α) | α∈R} is equal to 2. Consider the subset Γ_max:={α_i ∈Γ_a |m_i =m_max}of Γ_a, wherem_max:= max{m_i |0≤ i≤n}. Put Γ^∗_max:={α^∗_i |α_i∈Γ_max}.

Let (R, G) be a marked elliptic root system belonging toI. Then R^∨ :=

{α^∨ ∈F | α∈R} is also an elliptic root system belonging to I. Moreover it is known that the same space G defines a marking for R^∨. We call the pair (R^∨, G) the dual marked elliptic root system of (R, G).

4.2. Elliptic Dynkin diagrams. An elliptic Dynkin diagram Γ(R, G) for a marked elliptic root system (R, G) is a finite graph given by the following data:

(1) the vertex set of Γ(R, G) is Γ := Γ_a∪Γ^∗_max.

(2) two verticesα, β∈Γ are connected according to the following conditions:

e

α eβ ifI(α, β) =I(β, α) = 0, e

α eβ ifI(α, β^∨) =I(β, α^∨) =−1, e

α -^t eβ ifI(α, β^∨) =−μand I(β, α^∨) =−1 forμ= 2,3, e

α ∞ eβ

ifI(α, β^∨) =I(β, α^∨) =−2, e

α eβ IfI(α, β^∨) =I(β, α^∨) = 2.

Afterwards we use the following conventions:

e e = e -^μ e = e ^μ e forμ= 1, e -^μ e = e ^μ−1 e forμ=±2,±3.

The following theorem is due to K. Saito [S].

Theorem 4.2.1. The isomorphism classes of marked elliptic root sys- tems are completely classified by their elliptic Dynkin diagrams.

In Appendix, we will present a complete list of marked elliptic root systems (R, G) under the assumption that R_f is reduced.

By the above classification theorem we have the following lemma.

Lemma 4.2.2. The component Γ(R, G)\(Γ_max∪Γ^∗_max) = Γ_a\Γ_max is a disjoint union ofA-type diagrams, sayΓ(A_l1),· · ·,Γ(A_lr).

Forα_i ∈Γ_a, we set α^†_i :=k_αiα^∨_i. It is known that the setQ((R, G)_a) :=

⊕ⁿ_i=0Zα^†_i forms a root lattice of an irreducible reduced affine root system (R, G)_a with a basis{α_i^†}ⁿ_i=0. In order to describe the explicit type of (R, G)_a, we introduce a grouping of isomorphism classes of marked elliptic root systems due to K. Saito [S]:

(A) A^(1,1)_n (n≥1),D^(1,1)_n (n≥4), E_n^(1,1)(n= 6,7,8),

(B) B_n^(1,2)(n≥3),B_n^(2,2)(n≥2),C_n^(1,2)(n≥2),C_n^(2,2)(n≥3),F₄^(1,2),F₄^(2,2), G^(1,3)₂ ,G^(3,3)₂ ,

(C) B_n^(1,1)(n≥3),B_n^(2,1)(n≥2),C_n^(1,1)(n≥2),C_n^(2,1)(n≥3),F₄^(1,1),F₄^(2,1), G^(1,1)₂ ,G^(3,1)₂ ,

(D) A^(1,1)∗₁ ,B_n^(2,2)∗ (n≥2),C_n^(1,1)∗ (n≥2).

Theorem 4.2.3. If (R, G)belongs to the group A, B or C, we have

Q((R, G)_a)∼=

⎧⎪

⎨

⎪⎩

Q(R_a) =Q(R^∨_a), if(R, G)belongs to the group A, Q(R_a), if(R, G)belongs to the group B, Q(R^∨_a), if(R, G)belongs to the group C.

If(R, G)belongs to the group D, we have

Q((A^(1,1)∗₁ )_a)∼=Q(S(A₁)), Q((B_n^(2,2)∗)_a)∼=Q((C_n^(1,1)∗)_a)∼=Q(S(BC_n)).

If (R, G) belongs to the group A, B or C, there exists the irreducible reduced finite root system R⁽⁰⁾_f such that Q(R, G)_a is isomorphic to Q(S(R⁽⁰⁾_f )) or Q(S(R⁽⁰⁾_f )^∨). But in general, R⁽⁰⁾_f is not isomorphic toR_f.

4.3. Boundary side. Let us introduce the notion of the boundary side due to K. Saito and Takebayashi [ST]. For each pairα_i, α_j ∈Γ_a which are connected as _αib -^μ b_αjforμ= 2^±1,3^±1, it is known thatk(α_i, α_j) :=k_αi/k_αj is equal to either 1 orμ.

Definition 4.3.1. In the above setting, α_i is called the boundary side (or b-side for short) for the bond _αib -^μ b_αjwithμ= 2^±1,3^±1, ifk(α_i, α_j) = inf{1, μ}.

Remark. For the bond _αib -^μ b_αj forμ= 2^±1,3^±1, eitherα_iorα_j is a b-side.

4.4. Hyperbolic extension of elliptic Weyl groups. Let (R, G) be a marked elliptic root system. Consider the pair ( ˜F ,I) of a vector space ˜˜ F over R and a symmetric bilinear form ˜I on ˜F such that F is a 1-codimensional subspace of ˜F, ˜I|_F = I and rad( ˜I) = G. Such ( ˜F ,I) exists uniquely up˜ to isomorphisms. By the definition, we can regard R is a subset of ˜F. Let

˜s_α∈O( ˜F ,I) be the reflection with respect to˜ α∈R and W(R, G) the group

generated by all reflections ˜s_α(α∈R) which is called the hyperbolic extension ofW(R).

K. Saito and Takebayashi [ST] gave a presentation ofW(R, G) by genera- tors attached to the vertices of the elliptic Dynkin diagram Γ(R, G) and finitely many relations. In their presentation, in addition to the ordinary Coxeter rela- tions, new relations (so-called elliptic Coxeter relations) appeared. After [ST], Yamada [Y] gave a modification of K. Saito and Takebayashi’s presentation for one-codimensional cases. In this article, we generalize Yamada’s presentation ofW(R, G) for arbitrary marked elliptic root systems.

The following is K. Saito and Takebayashi’s presentation ofW(R, G).

Theorem 4.4.1 ([ST]). W(R, G)is isomorphic to the group with gen- eratorsr_α (α∈R)subject to the relations explained below.

For any subdiagrams ofΓ(R, G) isomorphic to the following list, we give relations attach to the diagrams in the following table.

(W0) α e r²_α= 1,

(W1-1) α e eβ (rαrβ)²= 1,

(W1-2) α e eβ (rαrβ)³= 1,

(W1-3) α e -^2±1 eβ (rαrβ)⁴= 1, (W1-4) α e -^3±1 eβ (rαrβ)⁶= 1,

(W2-1) (rαrβrα^∗rβ)³= 1, e

α^∗

e

α

eβ

HHHH

(W2-2)

(rβrαrα^∗)²= (rαrα^∗rβ)²,

(rβrαrα^∗)² commutes withrα, rα^∗, andrβ,

e

α^∗

e

α

eβ

HHHH

∞

(W2-3) rαrβ^∗rα=rα^∗rβrα^∗,

⇔rβrα^∗rβ=rβ^∗rαrβ^∗ under(W0)and(W1-2), e

α^∗ eβ^∗

e

α eβ

@@

@@

(W2-4) rαrα^∗rβrβ^∗ =rα^∗rβrβ^∗rα

=rβrβ^∗rαrα^∗ =rβ^∗rαrα^∗rβ, e

α^∗ eβ^∗

e

α eβ

@@

@@

∞

(W3-1)

(rαrβrα^∗rβ)²= 1, e

α^∗

e

α

eβ

HHHH HHHj

*

2^±

(W3-2) (rαrβrα^∗rβ)³= 1, (rαrβrα^∗rβrαrβ)² = 1, e

α^∗

e

α

eβ

HHHH HHHj

*

3^±

In the next diagram, we assume that α is b-side for the bond _α b -^μ b_{β for} μ= 2^±1,3^±1.

(W3-3)

rαrβ^∗rα=rα^∗rβrα. e

α^∗ eβ^∗

e

α eβ

-

-

@@

@@

@@@R

2^±1

In the next diagram, we assumeμ= 1,2^±1,3^±3.

(W4)

(rαrβrαrβ^∗rγrβ^∗)²= 1, (rαrβ^∗rαrβrγrβ^∗)²= 1.

e

α

βe^∗

e

β

eγ

HHHH

HHHH HHHj

*

μ

However there are exceptions. In the diagram (W2-4), there are four subdia- grams of type (W2-2). But, we do not assume the relations (W2-2) for these four subdiagrams. We only assume the relations(W2-4).

The relations (W2-1)∼(W4) are called elliptic Coxeter relations.

The following theorem is a generalization of Yamada’s presentation [Y].

Theorem 4.4.2. W(R, G)has another presentation with generatorsr_α (α∈R)subject to the relations explained below.

For any subdiagrams ofΓ(R, G) isomorphic to the following list, we give relations attach to the diagrams in the following table.

(E0) α e r²α= 1, (as same as(W0)),

(E1-1) α e eβ rαrβ=rβrα, (E1-2) α e eβ rαrβrα=rβrαrβ,

(E1-3) α e -^2±1 eβ rαrβrαrβ=rβrαrβrα,

(E1-4) α e -^3±1 eβ rαrβrαrβrαrβ=rβrαrβrαrβrα,

In the following diagrams, we always assume that α, β, γ ∈Γ_a. Forα ∈Γ_max, set x_α†=rαrα^∗.

(E2-1) rβx_α†rβx_α†=x_α†rβx_α†rβ, e

α^∗

e

α

eβ

HHHH

(E2-2)

rβx_α†rβx_α†=x_α†rβx_α†rβ,

rβx_α†rβx_α† commutes withrα, rα^∗, andrβ,

e

α^∗

e

α

eβ

HHHH

∞

(E2-3) x_β†x_α†=rβx_α†rβ,

⇔x_α†x_β†=rαx_β†rα under(E0)and(E1-2), e

α^∗ eβ^∗

e

α eβ

@@

@@