数理解析研究所講究録1086 巻 1999 年 57-70

57

Elliptic Root System and Elliptic Artin Group
Hiroshi Yamada
Kitami Institute of Technology
Kitami Hokkaido 090 Japan

Introduction

1

The concepts of elliptic root system, elliptic Dynkin diagram and elliptic Weyl group wereintroduced by K. Saito to describe the Milnor lattices and the flat structures of semi-universal
deformations for simply elliptic singularities $[10][11][12][13]$ .
, K.Saito and T.Takebayashi studied generators and relations of elliptic
Furthermore,
Weyl groups in terms of elliptic Dynkin diagrams (This presentation of elliptic Weyl group isa generalization of Coxeter system. See Theorem 2.1). In the paper, they also proposed thefollowing problems: find generators and relations of “elliptic Lie algebras”, “elliptic Hecke algebras” and elliptic Artin groups (the fundamental groups of the complements of the discriminant
for simply elliptic singularities) in terms of the elliptic Dynkin diagrams.
, K.Saito and D.Yosh\"u conIn [14], applying Borcherd’s cnstruction of vertex
structed the elliptic Lie algebras (which are isomorphic to the toroidal algebras [8]) and described generators and relations of them for homogeous elliptic Dynkin diagrams (This presentation is a generalization of Serre relations attached to the elliptic Dynkin diagrams. cf.
$[1],[17])$ .
In [18], H. van der Lek has given a presentation of the elliptic Artin groups (which he calls theextended Artin groups) using affine Dynkin diagrams. The aim of this note is to give anotherpresentation of elliptic Artin groups in terms of elliptic Dynkin diagrams. In our presentations,the numbers of generators and relations are less than his ones. Moreover, as a by-product, weshall define elliptic Hecke algebras (which are subalgebras of Cherednik’s double affine Hecke
algebras $[4],[5],[6])$ and construct finite dimensional irreducible representations of them.
Here,we briefly explain H. van der Lek’s description of the elliptic Artin groups. Let $C=$
be the Coxeter matrix determined
be an affine Cartan matrix andas follows:by$\mathrm{i}\mathrm{n}[15]$$\mathrm{a}\mathrm{l}\mathrm{g}\mathrm{e}\mathrm{b}\mathrm{r}\mathrm{a}\mathrm{s}[2][[3]$$M=(m_{i,j})_{0}\leq i,j\leq \mathrm{t}$$(c_{t\theta})0\leq i,j\leq\iota$$c_{i,j}$$m_{i,j}=2,3,4,6,$$\infty$$\dot{i}f$$c_{t,j}c_{j,i}=0,1,2,3,$ $\geq 4$,

respectively.

) Let
be an affine root system and $C(R_{a})$ the affine
Theorem 1.1 (H. van deris generated by. The elliptic Artin group $A(R_{a})$ associated with
Cartan Matrix oft,} which satisfy the following relations:{$\mathrm{L}\mathrm{e}\mathrm{k}[18]$$R_{a}$$R_{a}$$R_{a}$$s_{0},$$s_{1,\cdots,\iota,}s$

(A.1)
(A.2)

$t_{1},$$\cdots,$$t_{\iota}$$s_{i}s_{ji}S\cdots=s_{j}s_{ij}S\cdots$$t_{i}t_{j}=t_{j}t_{i}$each side$m_{i,j}$factors if$i\neq j$

58

(A.3)

$2r=-c_{j,i}$$s_{i}t_{j}t_{i}^{r}=t_{j}t_{i}^{r_{S_{i}}}$(A.4)$2r+1=-c_{j},i$$s_{i}t_{ji}t^{r}=ti^{S}jt^{r+1}i$

In this note, for simplicity, we shall only treat with elliptic Dynkin diagrams obtained byadding one vertex to affine Dynkin diagrams (see the appendix) and, for brevity, call themelliptic Dynkin diagrams hereafter.

2

Elliptic root systems and elliptic Weyl groups

We briefly explain elliptic root systems and elliptic Weyl groups following [12]. Let $F$ be avector space over with symmetric bilinear formof signature (, where(resp.
is the dimension of a maximal positive (resp. negative) definite subspace of $F$ and is thedimension of the radical of. Forsuch that, we define$\mathrm{R}$$(\cdot, \cdot)$$l_{+},$$l_{0,-)}l$$l_{+}$$l_{-})$$l_{0}$$\alpha\in F$$(\cdot, \cdot)$$(\alpha, \alpha)\neq 0$$\alpha^{\vee}=\frac{2}{(\alpha,\alpha)}\alpha$$w_{\alpha}(u)=u-(u, \alpha^{\vee})\alpha$,

for any$u\in F$.

Definition 2.1 A subset $R\subset F$ is called an elliptic root system, if the following conditions
are satisfied:(R.1)$(l_{+},$$l_{0},$$l_{-)=}(l, 2,0)$(R.2)

Let $Q(R)$ be the(R.3)

For any(R.4)$w_{\alpha}(R)=R$(R.5)$\alpha\in R,$

If-submodule of $F$ generated by R. Then,$(\alpha, \alpha)\neq 0$$Q(R)\otimes \mathrm{z}\mathrm{R}=F$

.

.

.

$(\alpha, \beta^{\vee})\in \mathrm{Z}$(R.6)$\mathrm{Z}$.

$R=R_{1}\cup R_{2}$for any$\alpha,$$\beta\in R$.

, then $R_{1}=\phi$ or$R_{2}=\phi$.

We call the Weyl group $W(R)$ associated with elliptic Weyl group of . Also K. Saito defined
an elliptic Dynkin diagram for an elliptic root system as follows: Letbe a l-dimensional
subspace of rad
which is defined over Q. Then, the quotient of by is an affine rootsystem. We fix a generator of the lattice $G\cap Rad((, ))$ . Note that the generator is unique
up to a choice of sign. We call $(R, G)$ the marked elliptic root system. For any
, put
$R$

$R$

$G$

$R$

$(\cdot, \cdot)$$R_{a}$

$a$

$G$

$a$

$\alpha\in R_{a}$$k( \alpha)=\inf\{k\in N|\alpha+k.

a \in R\}$

and$\alpha^{*}=\alpha+k(\alpha)\cdot a$$k(\alpha)$.

is called the counting of . Then we have the following proposition:$\alpha$59

Proposition 2.1 (K.Saito [12]) Let $(R, G)$ be a marked elliptic root system. Then we have
$R=\{\alpha+m\cdot k(\alpha)\cdot a|\alpha\in R_{a}, m\in Z\}$

Let$\Gamma_{a}=\{\alpha_{0}, \alpha_{1}, \cdots, \alpha\downarrow\}$$\{\alpha_{1}, \cdot\cdot’, \alpha_{l}\}$:.

such thatbe a basis ofsimple roots of finite root system$R_{a}$$\alpha_{0}=b-\Sigma\iota i=1ni\alpha_{i}$

$b$

of

$0$

:

imaginary root of$R_{a}$. The set of exponents of $(R, G)$ is defined by the union

and

for

$m_{\alpha}= \frac{(\alpha,\alpha)_{R}}{2\cdot k(\alpha)}\cdot n_{\alpha}$

where
2. Set$(\cdot, \cdot)_{R}$is a constant multiple of$(\cdot, \cdot)$$\alpha\in\Gamma_{a}$normalized such that inf$\{(\alpha, \alpha)_{R}|\alpha\in R\}$is equal to$\Gamma_{a,\max}=\{\alpha\in\Gamma_{a}|m_{\alpha}=\max\{m_{\beta}|\beta\in \mathrm{r}\}\}$

and$\Gamma_{a,\max}^{*}=\{\alpha*|\alpha\in\Gamma_{a},\max\}$.

Remark 2.1 In this note, we assume that the number of the verteces of$\Gamma_{a,\max}$

is equal to 1.

Definition 2.2 The elliptic Dynkin diagram $\Gamma(R, G)$ of the marked elliptic root system $(R, G)$and bonded by the following
is a finite graph generated by the set of verteces$\beta\in\Gamma(R, G)$ ,conditions: for$\Gamma(R, G)=\Gamma_{a}\cup\Gamma_{a,\max}*$$\alpha,$(1)$(\alpha, \beta^{\vee})=0$(2)$(\alpha, \beta^{\vee})=(\alpha^{\vee}, \beta)---1$(3)$(\alpha, \beta^{\vee})=-t,$ $(\alpha, \beta^{\vee})=-1$(4)$(\alpha, \beta^{\vee})=(\alpha^{\vee}, \beta)=2$$\alpha \mathrm{O}$$0\beta$$\alpha \mathrm{R}\beta$

for $t=2,3,4$$\alpha \mathrm{R}_{t}\beta$$\alpha \mathrm{O}=;\mathrm{O}\beta$

Let us define the equality$m(R, G)= \frac{\max\{m_{\alpha}|\alpha\in\Gamma(R,G)\}}{gcd\{m_{\alpha}|\alpha\in\Gamma_{a}\}}$

This number $m(R, G)$ plays the role of the Coxter nummber for the elliptic root system $[11],[12],[13],[15]$ .
Using elliptic Dynkin diagrams, K. Saito and T. Takebayashi [15] gave presentations of elliptic
Weyl groups as follows:
Theorem 2.1 (K. Saito and T. Takebayashi [15]) Let $(R, G)$ be a marked elliptic rootsystem andthe group defined by the following generators and relations:$\overline{W}(R, G)$

generators :$r_{\alpha}$$\alpha\in\Gamma(R, G)$

60

relations:(W.O)$r_{\alpha}2=1$$\alpha \mathrm{O}$(W.1.0)$(r_{\alpha}r_{\beta})^{2}=1$(W. 1.1)$(r_{\alpha}r_{\beta})^{3}=1$$\alpha\mapsto\beta$(W.1.2)$(r_{\alpha}r_{\beta})^{4}=1$$\alpha \mathrm{R}\beta$(W.1.3)$(r_{\alpha}r_{\beta})^{6}=1$$\alpha \mathrm{K}\beta$(W.2.1)$(r_{\alpha}r_{\beta}r_{\alpha}tr_{\beta})^{3}=1$(W.2.2)$(r_{\alpha}r_{\beta\alpha^{\mathrm{s}}}rr\beta)^{2}=1$(W.2.3)$(r_{\alpha}r_{\beta\alpha\beta}rtr)^{3}=1$(W.2.3)$(r_{\alpha}r_{\alpha}*r_{\beta})^{2}=(r_{\alpha}*r_{\beta}r_{\alpha})^{2}=(r_{\beta\alpha\alpha^{*)^{2}}}rr$(W.3)$(r_{\alpha}r_{\beta}r\alpha r_{\beta^{*}}r_{\gamma\beta}r*)^{2}=1$$\alpha \mathrm{O}$$0\beta$

$r$

for

and

$(r_{\alpha}r_{\beta}r_{\alpha^{*}}r\beta r\alpha r\beta)^{2}=1$

and$(r_{\alpha}r_{\beta}^{*}r\alpha r\beta r_{\gamma}r\beta)^{2}=1$$t=\mathit{1},\mathit{2},\mathit{3}$where the two relations are equivalent in the case of $t=1$ .

Define$\tilde{c}(R, G)$$=$$\prod_{i\in \mathrm{t}^{0}1,\cdot\cdot,\iota\}\backslash j}.r\alpha i.j\in J\prod r\alpha jr^{*}\alpha j$.

,

Then, the power $\tilde{c}(R, G)m(R,c)$ is a center of $\tilde{W}(R, G)$ and one has$\tilde{W}(R, G)/<\tilde{c}(R, G)m(R,c)>$$\cong$$W(R)$ .

(W.1.3) are well known as Coxeter relations, and the relations
Here the relations (W.1.0)$(W.2.1)\sim(W.3.t)$ are newly introduced relations due to the double bonds in the diagram. Let
hyperbolic elliptic Weyl groupus call them elliptic Coxeter relations and the groupof $W(R)$ (see $[12,[15]$ ).
$\sim$$\tilde{W}(R, G)$

Remark 2.2 In this note, we have assumed that the number of the vertces of,K. Saito and T. Takebayashi treated more general elliptic root systems.
to 1. In$\Gamma_{a,\max}$$f\mathit{1}\mathit{5}J$is equal61

3

Twisted Picard-Lefschetz formula

The Coxeter relations of elliptic Weyl groups were obtained by studying the monodromy
representations of simply elliptic singularities. To find generators and their relations of elliptic
Artin groups attached to elliptic Dynkin diagrams, we want to construct a certain kind of deformation of the monodromy representations of simply elliptic singularities. Fortunately, A. B.
has already studied -deformation of monodromy representations of isolated hypersurface singularities using the so called twisted Picard-Lefschetz formula (see also F.
.
andFirst we begin with explaining the classical Picard-Lefschetz formula.
be a polymonial mapping such that $f^{-1}(0)$ has a simply elliptic:Let$\mathrm{G}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{l}[7]$$\mathrm{q}$$\mathrm{P}\mathrm{h}\mathrm{a}\mathrm{m}[\mathfrak{g}]$$\mathrm{I}.\mathrm{S}\mathrm{h}\mathrm{i}\mathrm{m}\mathrm{a}\mathrm{d}\mathrm{a}[16])$$f$$(\mathrm{C}^{3}, \mathrm{O})arrow(\mathrm{C}, 0)$

singularity. Namely,$E_{6}^{(1,1)}$$E_{7}^{(1,1)}$

$\{$

:
:

$f(x, y, z)$$=$$f(x, y, z)$:$f(x,y, z)$$E_{8}^{(1,1)}$$z^{3}$$x^{3}$$+$$y^{3}$$+$$=$$x^{4}$$+$$y^{4}$$+$$z^{2}$$=$$x^{6}$$+$$y^{3}$$+$$z^{2}$.

Since a simply elliptic singilarity has a semi-universal deformation, there exists the following:
commutative diagram which is called a Hamiltonian system$\mathrm{i}\mathrm{n}[10]$$x=\mathrm{c}3_{\cross}\mathrm{c}\mu-1\Leftrightarrow_{\mathrm{C}}\mathrm{c}3\mathrm{F}\iota\cross \mathrm{C}^{\mu-1}\cross \mathrm{C}=Z$$pr_{l}\downarrow\sim^{\tilde{\pi}}f\downarrow \mathrm{P}^{\Gamma}\iota$$T=\mathrm{C}^{\mu-1}arrow \mathrm{C}^{\mu-1}\cross \mathrm{C}=S$
$\pi$

where
$\tilde{\pi},$$\{$$\pi,pr_{1},pr2$: natural projections,$F_{1}(x, y, z, t1, \cdots, t_{\mu-1})=(x, y, z, t_{1,1}\ldots, t-\mu’\hat{F}1(x, y, Z, t_{1-1}, \cdots,t_{\mu}))$,$\hat{F}_{1}(x, y, z, t_{1,\mu-}\ldots, t1)=f(X, y, Z)+\sum_{j1}\mu-1=t_{j}\phi j$and$\phi=pr_{1}\circ F_{1}$Here $\mu=l+2$ and$\{\phi_{j}\}_{j=1}^{\mu}$$deg(\phi_{j+1})\leq deg(\phi_{j})$is a$\mathrm{C}$:semi-universal deformation of .
$f$-basis of the Jacobi ring$\mathrm{C}[x, y, z]/(\partial\lrcorner\partial\lrcorner\partial x’\partial y’\partial z\lrcorner\partial)$

of

$f$

such that.

is a reduced
the discriminant of . The discriminant
be the critical set of andLetirreducible hypersurface in . Let $t’\in T=\mathrm{C}^{\mu-1}$ be a point which is not contained in the image. Setof the ramification locus of$\phi$$C_{\phi}$$D_{\phi}$$\phi$$D_{\phi}$$S$$\pi|_{D_{\phi}}$.

$L_{t’}=\{t’\}\cross \mathrm{C}\subset \mathrm{C}^{\mu-1}\cross \mathrm{C}=S$. We
with the discriminant
By choice of , there are exactly intersection points of, . A fibre $X_{p:}=\phi^{-1}(p_{i})$ has a singularity which is the ordinary
denote these points
is a 2-dimensional. Then the fibre
double point. Let. Hence, the only
manifold and homotopically isomorphic to a bouquet of copies of spherewhich is a free -module of rank .
is the groupnon-trivial homology group of$t$$D_{\phi}$$L_{t’}$$\mu$$p_{1},$$\cdots$$p_{\mu}$$X_{p_{0}}=\phi^{-1}(p\mathrm{o})$$p_{0}\in L_{t’}\backslash \{p_{1}, \cdots , p_{\mu}\}$$S^{2}$$\mu$$\mathrm{Z}$$X_{\mathrm{P}0}$$H_{2}(X_{\mathrm{p}0}, \mathrm{Z})$

$\mu$

62

The intersection numbers of cycles define a symmetric bilinear form $( , )$ on this module withsignature $(l, 2,0)(\mu=l+2)$ .
Next, we shall explain a relation between elliptic Dynkin diagram and vanishing cycles.
Choose a simple arc infrom to not passing through other . Then
$L_{t’}$$l_{i}$$p_{0}$$p_{i}$$p_{j}$$X_{p_{0}}\subset\phi^{-1}(l_{i})arrow X_{p:}$:contraction

induces the mapping$c_{i}$The kernel of this mapping is a
Kerne1by , i.e.
$(c_{t})$:$\mathrm{Z}$$H_{2}(x_{p_{0}}, \mathrm{Z})arrow H_{2}(X_{p_{\{}}, \mathrm{Z})$-submodule of$H_{2}(x_{p_{0}}, \mathrm{Z})$.

of rank 1. Denote a generator of$e_{i}$

Kerne1$(c_{t})=\mathrm{Z}e_{i}$.

It can be shown that ifare chosen in such a way that and intersect only at for$i\neq j$ , thenis a free -basis of
and $H_{2}(x_{p_{0}}, \mathrm{Z})=Q(R)$ . Furthermore
the intersection matrix with respect to this basis determines the elliptic Dynkin diagram (see$[10],[11])$ .
Now, we explain the classical Picard-Lefschetz formula. To each path , we as sociate an element
by going along fromto a point near , then turning counterclockwise
in a small circle aroundand then returning toalong . Then
is a set ofgenerators of). The mapping$l_{1},$$\cdots,$$\{e_{1}, \cdots, e_{\mu}\}$$l_{i}$$l_{\mu}$$l_{j}$$p_{0}$$H_{2}(x_{p0}, \mathrm{Z})$$\mathrm{Z}$$l_{i}$$\gamma_{i}\in\pi_{1}(L_{t^{l}},p_{0})$$l_{i}$$p_{0}$$p_{i}$$p_{i}$$l_{i}$$p_{0}$$\{\gamma_{1}, \cdots, \gamma_{\mu}\}$$\pi_{1(s\backslash D_{\phi,p_{0}}}$$\phi:\phi^{-1}(S\backslash D\phi)arrow S\backslash D_{\phi}$is the projection of a fibre bundle. Hence one gets a monodromy representation$\rho:\pi_{1(s\backslash \emptyset,p}D0)arrow Aut(H_{2}(X_{p0}, \mathrm{z}))$.

Finally, we can state the classicalformula:Classical Picard-Lefschetz formula (see [9])$\mathrm{P}\mathrm{i}\mathrm{C}\mathrm{a}\mathrm{r}\mathrm{d}-\mathrm{L}\mathrm{e}\mathrm{a}\mathbb{C}\mathrm{h}\mathrm{e}\mathrm{t}_{\mathrm{Z}}$for any$\rho(\gamma_{i})(\alpha)=\alpha-(\alpha, e_{i})e_{i}$$\alpha\in H_{2}(x_{p_{0}}, \mathrm{Z})$$(\dot{i}=1, \cdots, \mu=l+2)$$\hat{F}$Now, according to A. B. Givental [7], we explain the twisted Picard-Lefschetz formula. Define:by$Zarrow \mathrm{C}$$\hat{F}(x, y, z, t_{1}, \cdots, t_{\mu})=\hat{F}_{1}(x, y, Z, t1, \cdots, t_{\mu-}1)+t_{\mu}$and$\tilde{Z}=Z\backslash F^{-1}(0)$Since$\pi_{1}(\tilde{Z})\cong \mathrm{Z}$, for a complex number$q\in \mathrm{C}^{*}$.

, we can define a representation$\pi_{1}(\tilde{Z})arrow Aut(\mathrm{c})$:$1\vdash\neq q$.

This representation induces a local system
on . Define
and then:is a fibre bundle whose fibre is 3-dimensional complex manifold. For simplicity,$\tilde{Z}$$\mathcal{L}_{q}$$pr_{1}$$\tilde{Z}^{r}arrow S\backslash D_{\phi}$$\tilde{Z}^{r}=pr_{1}^{-1}(S\backslash D_{\emptyset})\cap\tilde{Z}$63

the restriction of
we also denote bymonodromy representation$\mathcal{L}_{q}$$\mathcal{L}_{q}$$\rho_{q}$:to the fibre$\tilde{Z}^{r}(p\mathrm{o})=pr_{1}^{-1}(p_{0})$$\pi_{1}(S\backslash D\phi)arrow Aut_{\mathrm{Z}[q,q^{-}}1](H3(\tilde{z}^{r}(p\mathrm{o}), \mathcal{L}_{q}))$. Then we get a.

This monodromy representation can be regarded as a -deformation of the classical one.
by
Denote the restriction of the mapping to$q$$\hat{F}$$\tilde{F}_{p0}$:$\tilde{Z}^{r}(p_{0})$$\tilde{Z}^{r}(p_{0})arrow \mathrm{C}^{*}$.

. Choose
has exactly critical values. We denote these points by
By choice of
to a point near the origin, then turning counterclockwise
going froma simple arc inalong the same way, and define a cycle
in a small circle around it, and finally returning toby carrying the vanishing cycle along . Then we obtain the following:$p_{0},\tilde{F}_{p_{\mathrm{O}}}$$\gamma_{i}’$$p_{1}’,$$\mu$$\cdots,p_{\mu}’$$\mathrm{C}^{*}$$p_{i}$$p_{i}’$$\delta_{i}\in H_{3}(\tilde{Z}^{r}(p\mathrm{o}), \mathcal{L})q$

Theorem 3.1 ($\mathrm{A}.\mathrm{B}$$e_{i}$. Givental [7]) (1)$\gamma_{i}’$$H_{3}(\tilde{Z}^{r}(p0), \mathcal{L})q=\oplus_{j=1}^{\mu}\mathrm{z}[q, q^{-1}]\delta_{j}$be an upper triangular matrix with diagonal elements 1 and-matrix $I_{q}=qV+^{t}$ V. Then one has
define a(2) Let$V$$(e_{i}, e_{j})$for $i<j$ and$\mu \mathrm{x}\mu$$\rho_{q}(\gamma_{i})(\delta_{j})=\delta_{jq_{i,j}i}-I\delta$.

(3) $(\rho_{q}(\gamma i)+q)(\rho q(\gamma_{i})-1)=0$ .

Applying this theorem to our problem, we obtain the following proposition:
Proposition 3.1 Set$g_{i}’=p_{q}(\gamma_{i})$$(i=1, \cdots, \mu=l+2)$ .

Then $g_{1}’’,$ satisfy the following relations:$\cdots,$ $g_{\mu}$(1)$(g_{i}’+q)(gi-\prime 1)=0$(2)$g_{i}’g_{j}’=g_{j}’g_{i}’$$g_{i}^{\prime\prime;}g_{j}g_{i}=g’jg’’igj$

Motivated by Proposition 3.2, we define the following group:$\alpha_{i\mathrm{O}}$$\alpha_{i\mathrm{O}}$$0^{\alpha_{j}}$$\alpha_{i\mapsto}\alpha_{j}$64Definition 4.1 Let $(R, G)$ be a marked elliptic root system and $\Gamma(R, G)$ be its elliptic Dynkin
diagram. Define a groupby the following generators and their relations:$\tilde{A}(R, G)$

generators:
relations:(E.1.0)(E.1.1)(E.1.2).
$\cdot$.

(E.1.3)(E.2.1)(E.2.2)(E.2.3)(E.2.4)(E.3)$\alpha\in\Gamma(R, G)$$g_{\alpha}$$0^{\beta}$$g_{\alpha}g_{\beta}=g_{\beta}g_{\alpha}$$\alpha \mathrm{O}$$g_{\alpha}g_{\beta}g_{\alpha}=g_{\beta}g_{\alpha}g_{\beta}$$\alpha\mapsto\beta$$g_{\alpha}g_{\beta}g\alpha g_{\beta}=g_{\beta g_{\alpha}}g_{\beta}g_{\alpha}$$\alpha \mathrm{r}_{\mathit{2}}\beta$$g_{\alpha}g_{\beta}g\alpha g\beta g_{\alpha}g\beta=g_{\beta}g_{\alpha}g\beta g_{\alpha}g_{\beta}g_{\alpha}$$Leu_{\alpha}=g_{\alpha}g_{\alpha^{\mathrm{s}}}$, then$\alpha \mathrm{r}_{\mathit{3}}\beta$$g_{\beta}t_{\alpha}g_{\beta\alpha}t=t_{\alpha}g_{\beta}t_{\alpha}g\beta$$g_{\beta}t_{\alpha}g_{\beta}g_{\alpha}=g_{\alpha}g_{\beta\alpha}tg_{\beta}$$g_{\beta}t_{\alpha}g_{\beta\alpha}t=t_{\alpha}g_{\beta}t_{\alpha}g\beta$

and$g_{\beta}t_{\alpha}g\beta g\alpha=g_{\alpha}^{*}g_{\beta}t_{\alpha}g_{\beta}$$g_{\beta}t_{\alpha}g_{\beta^{t}\alpha}=t_{\alpha}g_{\beta}t_{\alpha}g_{\beta}=g_{\alpha}^{*}g_{\beta^{t}\alpha}g\beta g_{\alpha}$$g_{\alpha}t_{\gamma}=t_{\gamma}g_{\alpha}$

where

for

and

$g_{\gamma}t_{\alpha}=t_{\alpha}g_{\gamma}$$t_{\gamma}=g_{\gamma}t_{\beta}g_{\gamma}t_{\beta}^{-1}$and$t_{\alpha}=g_{\alpha}t_{\beta}g_{\alpha}t_{\beta}^{-}1$$t=\mathit{1},\mathit{2},\mathit{3}$Here the relations (E.1.0)(E.1.3) are the same with (A.1) in Theorem 1.1 and the relations
are newly introduced ones due to the double bonds in the diagram $\Gamma(R, G)$ .
Now we consider the relation ofand the elliptic Artin group $A(R_{a})$ . To this purpose,
we introduce
as follows: For
is already defined by$\sim$$(\mathrm{E}.3)\sim(\mathrm{E}.5)$$\tilde{A}(R, G)$$t_{\alpha}\in\tilde{A}(R, G)$$\alpha_{0}\in\{\alpha|\alpha\in\Gamma_{a,\max}\},$$t_{\alpha 0}=g_{\alpha}0g_{\alpha 0}*$.

$t_{\alpha}0$

65

If

$\alpha_{k}\in\Gamma(R_{a})\backslash \mathrm{I}\alpha 1\alpha\in \mathrm{r}_{\mathrm{n}\mathrm{m}\mathrm{n}\mathrm{m}}\dagger$$\alpha_{1},$$\cdots,$

are arranged the following position

then we define$t_{\alpha j+1}=g_{\alpha}j+1t_{\alpha}jg_{\alpha}j+1t^{-1}\alpha j$.

inductively. Then we obtain the following lemma:
Lemma 4.1 Let $N(R, G)$ be a subgroup ofhas$\tilde{A}(R, G)$

generated by$\{t_{\alpha}|\alpha\in\Gamma(R, c)\}$. Then one

free abelian subgroup.

(1)$N(R, G)$ is a(2)$g_{\alpha}t_{\beta}=t_{\beta g}\alpha$$\alpha \mathrm{O}$$g_{\alpha}t_{\beta}g_{\alpha\alpha}=tt_{\beta}$$\alpha \mathrm{R}\beta$$g_{\beta}t_{\alpha\beta}t=t_{\alpha}t_{\beta}g_{\beta}$$0\beta$$\alpha \mathrm{r}_{\mathit{2}}\beta$$g\beta t\alpha t\beta g\beta=t\alpha t^{2}\beta$$\alpha \mathrm{r}_{\mathit{3}}\beta$$g\beta t_{\alpha}t_{\beta\beta}^{22}=t_{\alpha}tg_{\beta}$$\alpha \mathrm{r}_{\mathit{4}}\beta$(3) Setof$c(R, G)=R, \prod_{\alpha\in\Gamma(c)\backslash \{\alpha \mathrm{j}|j\in J\}}g_{\alpha}\prod_{j\alpha\in\{\alpha_{\mathrm{j}}|\in J\}}g\alpha g_{\alpha^{\mathrm{g}}}$$\tilde{A}(R, G)$, then the power $c(R, G)^{m(G}R,)$ is a centerand belongs to $N(R, G)$ . Especially, $c(R, G)m(R,c)$ is expressed by$c(R, G)m(R,G)= \prod_{)\alpha\in\Gamma(Ra}t_{\alpha^{\alpha}}n$are the coefficients of the imaginary root of the affine root system
where
Here $c(R, G)$ is called the Coxeter element of$n_{\alpha}$$R_{a}$.

$\tilde{A}(R, G)$

By Theoreml.1 and this Lemma 4.1, we obtain the following theorem:
the corresponding affine root
Theorem 4.1 Let $(R, G)$ be a marked elliptic root system andis isomorphic to the elliptic Artin group $A(R_{a})$ .
system. Then the group$R_{a}$$\tilde{A}(R, G)$

Therefore, we obtain generators and their relations of an elliptic Artin group associated with
an elliptic Dynkin diagram.
In $[4],[5],[6]$ , I.Cherednik defined the concept of double affine Hecke algebra and proved Macdonald’s inner product conjecture. We shall define the elliptic Hecke algebra which can beproved to be a subalgebra of his algebra.

66, the elliptic Hecke
Definition 4.2 Let $(R, G)$ be a marked elliptic root system. For
algebra $H_{q}(R, G)$ associated with $(R, G)$ is the quotient of the group algebrabythe relations$(g_{\alpha}+q)(g_{\alpha}-1)=0$(E.0)
for $\alpha\in\Gamma(R, G)$ .
$q\in \mathrm{C}^{*}$$\mathrm{C}(q)[\tilde{A}(R, G)]$

where$\mathrm{C}(q)$is the quotient field of$\mathrm{C}[q, q^{-}]1$.

Remark 4.1 (1) When $q=1$ , the relations (E.1.0).
Coxeter relations

are equivalent to the elliptic$\sim(E.\mathit{3})$$(W.\mathit{1}.\mathit{0})\sim(W.\mathit{3})$(2) Cherednik’s double affine Hecke algebra contains two parameters. In the elliptic Heckeand the power of the Coxeter
algebra, two parameters appear from the local system$c(R,c)^{m()}R,G$
element,.
$\mathcal{L}_{q}$$T$

Let $C(R, G)$ be the Cartan matrix corresponding to an elliptic Dynkin diagram $\Gamma(R, G)$ andbe the upper triangular matrix with diagonal elements 1 such that.

$C(R, G)=T+^{\iota}\tau$Difine$\mu\cross\mu$-matrix$C_{q}(R, G)=q\cdot T+^{t}T$

whrere$\mu=\mathrm{t}\mathrm{h}\mathrm{e}$,number of vertces of $\Gamma(R, G)$ .

is equal to 1.
Note that we have assume that the number of vertces of$\alpha\in\Gamma(R,G)$ , define the element, for any
On the vector spaceof $Aut(V(R, c))$ as follows : for any $\beta\in\Gamma(R, G)$ ,$\Gamma_{a,\max}$$V(R, G)=\oplus_{\alpha\in\Gamma()}R,c\mathrm{c}(q)\alpha$$A_{\alpha}(\beta)=\beta-c_{q(}R,$

where$C_{q}(R, c)_{\alpha,\beta}$is$(\alpha, \beta)$$c)\alpha,\beta.\alpha$$A_{\alpha}$,-component of $C_{q}(R, G)$ . Then we obtain the following proposition:

Proposition 4.1 Let $(R, G)$ be a marked elliptic root system such that the number of vertces
is equal to 1. Then one hasof$\Gamma_{a,\max}$(1)$p_{q}$; $\tilde{A}(R, G)arrow Aut(V(R, G))$$\rho_{q}(g_{\alpha})=A_{\alpha}$

is a fine dimensional irreducible representation of$\tilde{A}(R, G)$

over$\mathrm{C}(q)$.

(2) The above representation induces the follwing commutative diagram:$H_{q}(\mathit{1}\{,$$G1\cdot$

Especially, one obtains a finite dimensional irreducible representation of $H_{j}(R, c)$ .

67

References[1] S.Berman and R.V.Moody: Lie algebras graded by finite root systems and intersection matrix
algebras of Slodowy.
Invent. Math., 108, (1992). 323-347[2] R. Borcherds: Vertex algebras, $Kac$ -Moody algebras and the Monster.
Proc. Natl. Acad. Sci. USA 83, (1986). 3068-3071[3] R. Borcherds: Monstrous moonshine and monstrous Lie superalgebras.
Invent. Math. 109. (1992), 405-444[4] I. Cherednik: Double Affine Hecke Algebras, Knizhnik-Zamolodchikov Equations, and Macdonald’s Operators.
International Math. Research Notice No.9. (1992), 171-179[5] I. Cherednik: Double Affine Hecke Algebras and Macdonald’s conjecture.
Ann. Math. 141. (1995), 191-216[6] I.Cherednik: Macdonald’s evaluation conjectures and difference Fourier transform.
Invent. Math.122. (1995), 119-145[7] A. B. Givental: Twisted Picard-Lefschetz formulas.
Functional Anal. Appl. vol22. No.l (1987), 10-18[8] R.V.Moody, S. Eswara Rao and T. Yokonuma: Toroidal Lie algebras and vertex representation.
Geom. Dedicata. 35 (1990), 283-307[9] F. Pham: Formules de Picard-Lefschetz g\’en\’eralis\’ees et ramification de int\’egrales,
Bull. Soc.math.France, 93 (1965),333-367[10] K. Saito: Einfach Elliptische Singularitaten,
Invent. Math. 23 (1974), 289-325[11] K. Saito: Period Mapping Associated to a Primitive Form,
Publ.RIMS, Kyoto Univ. 19 (1983), 1231-1264[12] K. Saito: Extended Affine Root Systems I.
Publ. RIMS,Kyoto Univ.21 No.l (1985), 75-179[13] K. Saito: Extended Affine Root Systems II (Flat Invariants),
Publ. RIMS,Kyoto Univ.26 No.1 (1990), 15-78[14] K.Saito and D.Yoshii: Extended affine Root System IV (Simply Laced Elliptic Lie Algebras),
preprint (1998)[15] K. Saito and T. Takebayashi: Extended Affine Root Systems III (Elliptic Weyl Groups).
Publ. RIMS,Kyoto Univ.33 No.2 (1997), 301-329

68[16] I. Shimada: Picard-Lefschetz theory for the universal covering of complements to affine
hypersurfaces.
to appear in Publ. RIMS, Kyoto Univ.
[17] P.Slodowy: Beyond $Kac$ -Moody algebras and inside. Can. Mth. Soc. Proc., 5 (1986), 117122[18] H. van der Lek: Extended Artin Groups.
Proc. of symp. in Pure Math. vo140 (1983), Part 2, 117-121

69$A_{1}^{(1,1)}*$$Bc_{1}^{\mathrm{t}2,1)}$$BC_{1}^{(2,4)}$

70