Double affine braid groups and Hecke algebras of classical type

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 12 March 2012

Introduction

I thank M. Noumi for providing me with a copy of his article [N95].

Type $(C_n^{\vee },C_n),$ for $n\ge 2$

Use a graphical notation for relations so that $$\begin{array}{rcl}\begin{array}{c} gi gj \end{array}& means& g_i{g}_j=g_j{g}_i,\\ \begin{array}{c} gi gj \end{array}& means& g_i{g}_j{g}_i=g_j{g}_i{g}_j, \; and\\ \begin{array}{c} gi gj \end{array}& means& g_i{g}_j{g}_i{g}_j=g_j{g}_i{g}_j{g}_i.\end{array}$$

The double affine braid group

The double affine braid group is the group ${\widetilde{𝔅}}_n$ is generated by $q^{\frac{1}{2}},T_1,...,T_n$ and $T_0,T_0^{\prime },T_0^{\vee }$ with relations $$\begin{array}{ll}{q}^{\frac{1}{2}}\in Z\left({\widetilde{𝔅}}_n\right), T_0^{\vee }{T}_1^{-1}{T}_0{T}_1=T_1^{-1}{T}_0{T}_1{T}_0^{\vee }, T_0^{-1}{T}_1{\left(T_0^{\vee }\right)}^{-1}{T}_1^{-1}=T_1{\left(T_0^{\vee }\right)}^{-1}{T}_1^{-1}{T}_0^{-1},& (Daff\; 2.1)\end{array}$$ $$\begin{array}{ll}\begin{array}{r} T0 T1 T2 Tn-2 Tn-1 Tn T0∨ T1 T2 Tn-2 Tn-1 Tn T0′ T1 T2 Tn-2 Tn-1 Tn \end{array}& (Daff\; 2.2)\end{array}$$ There are redundant relations in this presentation of ${\widetilde{𝔅}}_n.$ In particular, the second relation in (Daff 2.1) implies $T_1{T}_0^{\vee }{T}_1^{-1}{T}_0=T_0{T}_1{T}_0^{-1}{T}_1,$ giving that $T_0$ commutes with $T_1{T}_0^{\vee }{T}_1^{-1}$ and thus that $T_0^{-1}$ commutes with $T_1{\left(T_0^{\vee }\right)}^{-1}{T}_1^{-1}.$ But this is the last relation in (Daff 2.1).

The elements of the braid group on $n+3$ strands given by $$\begin{array}{ccc}\begin{array}{rcl}{T}_0& =& \begin{array}{c} \end{array}\end{array}& and& \begin{array}{rcl}{T}_n& =& \begin{array}{c} \end{array}\end{array}\\ \begin{array}{rcl}{T}_0^{\vee }& =& \begin{array}{c} \end{array}\end{array}& and& \begin{array}{rcl}{T}_i& =& \begin{array}{c} i i+1 \end{array},\end{array}\end{array}$$ satisfy the relations in (Daff 2.1) and (Daff 2.2).

For $j= 1,...,n$ define $Y^{{\epsilon }_j^{\vee }}$ and $X^{{\epsilon }_j}$ in ${\widetilde{𝔅}}_n$ by $$\begin{array}{ll}\begin{array}{rclcrcl}{Y}^{{\epsilon }_1^{\vee }}& =& T_0{T}_1\cdots T_n\cdots T_1& and& Y^{{\epsilon }_{j+1}^{\vee }}& =& T_j^{-1}{Y}^{{\epsilon }_j^{\vee }}{T}_j^{-1},\\ X^{{\epsilon }_1}& =& {\left(T_0^{\vee }\right)}^{-1}{T}_1^{-1}\cdots T_n^{-1}\cdots T_1^{-1}& and& X^{{\epsilon }_{j+1}}& =& T_j{X}^{{\epsilon }_j}{T}_j,\end{array}& (Daff\; 2.3)\end{array}$$ for $j= 1,...,n-1 .$ Pictorially, $$Y^{{\epsilon }_j^{\vee }}=\begin{array}{c} j \end{array}and{X}^{{\epsilon }_j}=\begin{array}{c} j \end{array}.$$ The pictorial viewpoint makes it straightforward to verify that the subgroups $$X=⟨X^{{\epsilon }_1},...,X^{{\epsilon }_n}⟩andY=⟨Y^{{\epsilon }_1^{\vee }},...,Y^{{\epsilon }_n^{\vee }}⟩are\; abelian.$$

In ${\widetilde{𝔅}}_n$ let $T_{s_{{\epsilon }_1}}=T_1\cdots T_n\cdots T_1$ and $$\begin{array}{ll}{T}_0^{\prime }=q^{-\frac{1}{2}}{X}^{{\epsilon }_1}{T}_0^{-1}=q^{-\frac{1}{2}}{\left(T_0^{\vee }\right)}^{-1}{T}_1^{-1}\cdots T_n^{-1}\cdots T_1^{-1}{T}_0^{-1}=q^{-\frac{1}{2}}{\left(T_0^{\vee }\right)}^{-1}{Y}^{-{\epsilon }_1^{\vee }}& (Daff\; 2.4)\end{array}$$ DRAW A PICTURE OF THIS ELEMENT? Then, with $T_{s_{{\epsilon }_1}}=T_1\cdots T_n\cdots T_1,$ $$\begin{array}{r} T0′ T1 T2 Tn-2 Tn-1 Tn \end{array}and{T}_0{T}_0^{\prime }{T}_0^{\vee }{T}_{s_{{\epsilon }_1}}=q^{-\frac{1}{2}},$$ since

The affine braid groups ${𝔅}_n,{𝔅}_n^{\prime }$ and ${𝔅}_n^{\vee }$ are the subgroups of ${\widetilde{𝔅}}_n$ given by $${𝔅}_n=⟨T_0,T_1,...,T_n⟩,{𝔅}_n^{\prime }=⟨T_0^{\prime },T_1,...,T_n⟩and{𝔅}_n^{\vee }=⟨T_0^{\vee },T_1,...,T_n^{\vee }⟩.$$

1. The double affine braid group ${\widetilde{𝔅}}_n$ is generated by $q^{\frac{1}{2}},$ the affine braid group ${𝔅}_n^{\vee },$ and the abelian group $Y$ with additional relations $$q^{\frac{1}{2}}\in Z\left({\widetilde{𝔅}}_n\right),Y^{{\epsilon }_{i+1}^{\vee }}=T_i^{-1}{Y}^{{\epsilon }_i^{\vee }}{T}_i^{-1},for\; i=1,...,n-1,$$ and $$\begin{array}{ll}\begin{array}{r} ⋯ ⋯ Yε1∨ Yε2∨ Yεj-1∨ Tj∨ Yεj+2∨ Yεn-1∨ Yεn∨ \end{array} \; for\; j=0,...,n.& (Daff\; 2.5)\end{array}$$
2. The double affine braid group ${\widetilde{𝔅}}_n$ is generated by $q^{\frac{1}{2}},$ the affine braid group ${𝔅}_n,$ and the abelian group $X$ with additional relations $q^{\frac{1}{2}}\in Z\left({\widetilde{𝔅}}_n\right),$ and $$\begin{array}{ll}\begin{array}{r} ⋯ ⋯ Xε1 Xε2 Xεj-1 Tj Xεj+2 Xεn-1 Xεn \end{array} \; for\; j=0,...,n.& (Daff\; 2.6)\end{array}$$

		Proof.
	(Daff 2.1) $\Rightarrow$ (a): The relations (Daff 2.5) for $j\ne 0$ follow easily from the pictorial expression for $Y^{{\epsilon }_j^{\vee }}$ and the additional relations (Daff 2.5) for $j=0$ follow from the definition of $Y^{{\epsilon }_j^{\vee }}$ and the relation $T_0^{\vee }{T}_1^{-1}{T}_0{T}_1=T_1^{-1}{T}_0{T}_1{T}_0^{\vee }.$ (a) $\Rightarrow$ (Daff 2.1): Since $$T_0=Y^{{\epsilon }_1^{\vee }}{T}_1^{-1}\cdots T_{n-1}^{-1}{T}_n^{-1}{T}_{n-1}^{-1}\cdots T_1^{-1},$$ the generators in (Daff 2.1) can be written in terms of the generators in (a). We need to show that the relations (Daff 2.5) imply $$T_0{T}_1{T}_0{T}_1=T_1{T}_0{T}_1{T}_0,T_0^{\vee }{T}_1^{-1}{T}_0{T}_1=T_1^{-1}{T}_0{T}_1{T}_0^{\vee },and{T}_0{T}_i=T_i{T}_0,for\; i=2,3,...,n.$$ Pictorially, $$T_{s_{\phi }}=\begin{array}{c} \end{array}=T_1\cdots T_n\cdots T_1,$$ so that $T_i{T}_{s_{\phi }}^{-1}=T_{s_{\phi }}^{-1}{T}_i,$ for $i=2,3,...,n.$ Thus, $$T_0{T}_i=Y^{{\epsilon }_1^{\vee }}{T}_{s_{\phi }}^{-1}{T}_i=Y^{{\epsilon }_1^{\vee }}{T}_i{T}_{s_{\phi }}^{-1}=T_i{Y}^{{\epsilon }_1^{\vee }}{T}_{s_{\phi }}^{-1}=T_i{T}_0,for\; i=2,...,n.$$ Using $T_0^{\vee }{Y}^{{\epsilon }_2^{\vee }}=Y^{{\epsilon }_2^{\vee }}{T}_0^{\vee }, Y^{{\epsilon }_2^{\vee }}=T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}$ and $T_0{T}_i=T_i{T}_0$ for $i=2,...,n,$ $$\begin{array}{rcl}{T}_0^{\vee }{T}_1^{-1}{T}_0{T}_1& =& T_0^{\vee }{T}_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}\cdots T_n^{-1}\cdots T_1^{-1}{T}_1\\ & =& T_0^{\vee }{Y}^{{\epsilon }_2^{\vee }}{T}_2^{-1}\cdots T_n^{-1}\cdots T_2^{-1}\\ & =& Y^{{\epsilon }_2^{\vee }}{T}_0^{\vee }{T}_2^{-1}\cdots T_n^{-1}\cdots T_2^{-1}\\ & =& T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}{T}_2^{-1}\cdots T_n^{-1}\cdots T_2^{-1}{T}_0^{\vee }\\ & =& T_1^{-1}{T}_0{T}_1{T}_0^{\vee }.\end{array}$$ Since $Y^{{\epsilon }_1^{\vee }}=T_0{T}_1\cdots T_n\cdots T_1$ and $Y^{{\epsilon }_2^{\vee }}=T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}=T_1^{-1}{T}_0{T}_1\cdots T_n\cdots T_2$ we have $$\begin{array}{rcl}{Y}^{{\epsilon }_1^{\vee }+{\epsilon }_2^{\vee }}& =& T_0{T}_1\cdots T_n\cdots T_2{T}_0{T}_1\cdots T_n\cdots T_2\\ & =& T_0{T}_1{T}_0{T}_2\cdots T_n\cdots T_2{T}_1\cdots T_n\cdots T_2.\end{array}$$ Then $T_1^{-1}{Y}^{{\epsilon }_1^{\vee }+{\epsilon }_2^{\vee }}=T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{Y}^{{\epsilon }_2^{\vee }}=T_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}=Y^{{\epsilon }_2^{\vee }}{Y}^{{\epsilon }_1^{\vee }}{T}_1^{-1}=Y^{{\epsilon }_2^{\vee }+{\epsilon }_1^{\vee }}{T}_1^{-1}$ gives $$\begin{array}{rcl}{T}_1{Y}^{{\epsilon }_1^{\vee }+{\epsilon }_2^{\vee }}& =& T_1{T}_0{T}_1{T}_0{T}_2\cdots T_n\cdots T_2{T}_1\cdots T_n\cdots T_2\\ & =& Y^{{\epsilon }_1^{\vee }+{\epsilon }_2^{\vee }}{T}_1\\ & =& T_0{T}_1{T}_0{T}_2\cdots T_n\cdots T_2{T}_{s_{\phi }}\\ & =& T_0{T}_1{T}_0{T}_{s_{\phi }}{T}_2\cdots T_n\cdots T_2\\ & =& T_0{T}_1{T}_0{T}_1{T}_2\cdots T_n\cdots T_2{T}_1\cdots T_n\cdots T_2,\end{array}$$ since $T_{s_{\phi }}{T}_i=T_i{T}_{s_{\phi }}$ for $i=2,...,n.$ thus $T_1{T}_0{T}_1{T}_0=T_0{T}_1{T}_0{T}_1.$ (b) Since $\iota \left(Y^{{\epsilon }_i^{\vee }}\right)=X^{{\epsilon }_i},$ applying the automorphism $\iota$ to the presentation in (a) gives the presentation in (b). $\square$

Letting (see [M03, (1.4.3)]) $${\alpha }_n=2{\epsilon }_{n}and{\alpha }_i={\epsilon }_i-{\epsilon }_{i+1},for\; i=1,...,n-1,$$ the relations (Daff 2.5) are equivalent to $Y$ being abelian and $$T_i{Y}^{{\lambda }^{\vee }}=Y^{s_i{\lambda }^{\vee }}{T}_i,if\; ⟨{\lambda }^{\vee },{\alpha }_i⟩=0,and{T}_i^{-1}{Y}^{{\lambda }^{\vee }}{T}_i^{-1}=Y^{s_i{\lambda }^{\vee }}, \; if\; ⟨{\lambda }^{\vee },{\alpha }_i⟩=1.$$ The last relation is vacuous for $i=n$ since $⟨{\lambda }^{\vee },2{\epsilon }_n⟩\in 2\mathbb{Z}$ for all ${\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}.$

The presentation of the double affine braid group given in (???) is implicit in [H06, ???] and quite explicit in [?, ???].

Automorphisms of the double affine braid group

(Duality) [M03, (3.5.1)] There is an involutive automorphism $\iota :{\widetilde{𝔅}}_n\to {\widetilde{𝔅}}_n$ given by $$\iota \left(q^{\frac{1}{2}}\right)=q^{-\frac{1}{2}},\iota \left(T_0\right)={\left(T_0^{\vee }\right)}^{-1},\iota \left(T_0^{\vee }\right)=T_0^{-1},and\iota \left(T_i\right)=T_i^{-1} \; for\; i=1,...,n.$$

		Proof.
	The involution $\iota$ fixes the first relation in (Daff 2.1), switches the second and third relations in (Daff 2.1), and switches the relations in (Daff 2.2). $\square$

Note that $$\iota \left(Y^{{\epsilon }_i^{\vee }}\right)=X^{{\epsilon }_i},for\; i=1,...,n.$$

The double affine Weyl group

Type $(C_n^{\vee },C_n)$

Let $${𝔥}_{\mathbb{Z}}=\sum _{i=1}^n\mathbb{Z}{\epsilon }_i^{\vee }and{𝔥}_{\mathbb{Z}}^{*}=\sum _{i=1}^n\mathbb{Z}{\epsilon }_i,with⟨{\epsilon }_i^{\vee },{\epsilon }_j⟩={\delta }_{ij}.$$ The Weyl group $W_0$ is generated by $s_1,...,s_n$ with $$\begin{array}{ll}\begin{array}{r} s1 s2 sn-2 sn-1 sn \end{array}and{s}_i^2=1,& (Daff\; 3.1)\end{array}$$ for $i=1,...,n.$ The group $W_0$ acts on ${𝔥}_{\mathbb{Z}}$ by $$\begin{array}{ll}\begin{array}{rl}{s}_i{\epsilon }_i^{\vee }={\epsilon }_{i+1}^{\vee }, s_i{\epsilon }_{i+1}^{\vee }={\epsilon }_i^{\vee },& for\; i=1,...,n-1 \; and\; j\ne i,i+1,\\ and{s}_n{\epsilon }_n^{\vee }=-{\epsilon }_n^{\vee },& s_n{\epsilon }_j^{\vee }={\epsilon }_j^{\vee },for\; j\ne n.\end{array}& (Daff\; 3.2)\end{array}$$ Then $W_0$ acts on ${𝔥}_{\mathbb{Z}}^{*}$ by setting $$\begin{array}{ll}⟨w\mu ,{\lambda }^{\vee }⟩=⟨\mu ,w^{-1}{\lambda }^{\vee }⟩,for\; w\in W_0,\mu \in {𝔥}_{\mathbb{Z}}^{*},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}.& (Daff\; 3.3)\end{array}$$

The double affine Weyl group is $$\begin{array}{ll}\tilde{W}=\left\{q^{k/2}{X}^{\mu }w{Y}^{{\lambda }^{\vee }} \right| k\in \mathbb{Z},w\in W_0,\mu \in {𝔥}_{\mathbb{Z}}^{*},{\lambda }^{\vee }\in {𝔥}_{\mathbb{Z}}\}& (Daff\; 3.4)\end{array}$$ with $$\begin{array}{ll}{q}^{\frac{1}{2}}\in Z\left(\tilde{W}\right),w{X}^{\mu }=X^{w\mu }w,w{Y}^{{\lambda }^{\vee }}=Y^{w{\lambda }^{\vee }}w,& (Daff\; 3.5)\end{array}$$ $$\begin{array}{ll}{X}^{\mu }{X}^{\nu }=X^{\mu +\nu },Y^{{\lambda }^{\vee }}{Y}^{{\sigma }^{\vee }}=Y^{{\lambda }^{\vee }+{\sigma }^{\vee }},and\; X^{\mu }{Y}^{{\lambda }^{\vee }}=q^{⟨\mu ,{\lambda }^{\vee }⟩}{Y}^{{\lambda }^{\vee }}{X}^{\mu },& (Daff\; 3.6)\end{array}$$ for $\mu ,\nu \in {𝔥}_{\mathbb{Z}}^{*},{\lambda }^{\vee },{\sigma }^{\vee }\in {𝔥}_{\mathbb{Z}},w\in W_0.$

Noting that $$\begin{array}{ll}q=X^{{\epsilon }_1}{s}_{{\epsilon }_1}{Y}^{-{\epsilon }_1^{\vee }}{X}^{{\epsilon }_1}{s}_{{\epsilon }_1}{Y}^{-{\epsilon }_1^{\vee }} \; where\; s_{{\epsilon }_1}=s_1{s}_2\cdots s_{n-1}{s}_n{s}_{n-1}\cdots s_2{s}_1,& (Daff\; 3.7)\end{array}$$ define $$\begin{array}{ll}{s}_0=s_{{\epsilon }_1}{Y}^{-{\epsilon }_1^{\vee }},s_0^{\vee }=X^{{\epsilon }_1}{s}_{{\epsilon }_1},s_0^{\prime }=q^{-\frac{1}{2}}{X}^{{\epsilon }_1}{s}_{{\epsilon }_1}{Y}^{-{\epsilon }_1^{\vee }}.& (Daff\; 3.8)\end{array}$$ The following proposition shows that $\tilde{W}$ is a quotient of that double affine braid group ${\widetilde{𝔅}}_n.$

The group $\tilde{W}$ is presented by generators $q^{\frac{1}{2}},s_0^{\vee },s_0,s_1,...,s_n$ and relations $$\begin{array}{ll}\begin{array}{rcl}{q}^{\frac{1}{2}}\in Z\left(\tilde{W}\right), {\left(s_0^{\prime }\right)}^2=1,& and& s_i^2=1, \; for\; i=0,...,n,\\ \begin{array}{c} s0 s1 s2 sn-2 sn-1 sn s0∨ s1 s2 sn-2 sn-1 sn \end{array}& and& s_0^{\vee }{s}_1{s}_0{s}_1=s_1{s}_0{s}_1{s}_0^{\vee }.\end{array}& (Daff\; 3.9)\end{array}$$

		Proof.
	With definitions of $s_0,s_0^{\vee }$ and $s_0^{\prime }$ as in (Daff 3.8) it is straightforward to check that the relations of Proposition 3.1 hold in $\tilde{W}.$ Since $$\begin{array}{ll}\begin{array}{rcl}{X}^{{\epsilon }_1}=s_0^{\vee }{s}_1\cdots s_n\cdots s_1,& and& X^{{\epsilon }_{j+1}}=s_j{X}^{{\epsilon }_j}{s}_j,and\\ Y^{{\epsilon }_1^{\vee }}=s_0{s}_1\cdots s_n\cdots s_1,& and& Y^{{\epsilon }_{j+1}^{\vee }}=s_j{Y}^{{\epsilon }_j^{\vee }}{s}_j,\end{array}& (Daff\; 3.10)\end{array}$$ the generators of $\tilde{W}$ can be written in terms of $q^{\frac{1}{2}},s_0^{\vee },s_0,s_1,...,s_n .$ Theorem 2.1(a) shows that the group presented in Proposition 3.1 satisfies the relations $$\begin{array}{ll}{q}^{\frac{1}{2}}\in Z\left({\widetilde{𝔅}}_n\right),Y^{{\epsilon }_i^{\vee }}{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{Y}^{{\epsilon }_i^{\vee }},for\; 1\le i,j\le n,& (Daff\; 3.11)\end{array}$$ $$\begin{array}{rll}{s}_n{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{s}_n, \; for\; j=1,...n-1,& s_0^{\vee }{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{s}_0^{\vee }, \; for\; j=2,...,n,& (Daff\; 3.12)\\ Y^{{\epsilon }_{i+1}^{\vee }}=s_i{Y}^{{\epsilon }_i^{\vee }}, \; and\; s_i{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{s}_i,& for\; i=1,...,n-1 \; and\; j\ne i,i+1.& (Daff\; 3.13)\end{array}$$ Similarly Theorem 2.1(b) shows that the group presented in Proposition 3.1 satisfies the relations $$\begin{array}{ll}{q}^{\frac{1}{2}}\in Z\left({\widetilde{𝔅}}_n\right),X^{{\epsilon }_i}{X}^{{\epsilon }_j}=X^{{\epsilon }_j}{X}^{{\epsilon }_i},for\; 1\le i,j\le n,& (Daff\; 3.14)\end{array}$$ $$\begin{array}{rll}{s}_n{X}^{{\epsilon }_j}=X^{{\epsilon }_j}{s}_n, \; for\; j=1,...,n-1,& s_0{X}^{{\epsilon }_j}=X^{{\epsilon }_j}{s}_0, \; for\; j=2,...,n.& (Daff\; 3.15)\\ X^{{\epsilon }_{i+1}}=s_i{X}^{{\epsilon }_i}{s}_i, \; and\; s_i{X}^{{\epsilon }_j}=X^{{\epsilon }_j}{s}_i,& for\; i=1,...,n-1 \; and\; j\ne i,i+1.& (Daff\; 3.16)\end{array}$$ The relations in (Daff 3.11) and (Daff 3.14) imply the first relation in (Daff 3.5) and the first two relations of (Daff 3.6). The relation $s_0^2=1$ gives $s_{{\epsilon }_1}{Y}^{-{\epsilon }_1^{\vee }}=Y^{{\epsilon }_1^{\vee }}{s}_{{\epsilon }_1}$ which implies $$s_n{Y}^{-{\epsilon }_n^{\vee }}=Y^{{\epsilon }_n^{\vee }}{s}_n.Similarly\; s_n{X}^{{\epsilon }_n}=X^{-{\epsilon }_n}{s}_n$$ follows from ${\left(s_0^{\vee }\right)}^2=1.$ In combination with the relations in (Daff 3.12), (Daff 3.13), (Daff 3.15), (Daff 3.16) these imply the last two relations in (Daff 3.5). The relation $s_0^{\vee }{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{s}_0^{\vee }$ gives $X^{{\epsilon }_1}{s}_{{\epsilon }_1}{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{X}^{{\epsilon }_1}{s}_{{\epsilon }_1}$ so that, $$for\; j=2,...,n, X^{{\epsilon }_1}{Y}^{{\epsilon }_j^{\vee }}=Y^{{\epsilon }_j^{\vee }}{X}^{{\epsilon }_1};similarly\; Y^{{\epsilon }_1^{\vee }}{X}^{{\epsilon }_j}=X^{{\epsilon }_j}{Y}^{{\epsilon }_1^{\vee }}$$ follows from $s_0{X}^{{\epsilon }_j}=X^{{\epsilon }_j}{s}_0,$ for $j=2,...,n.$ Finally, ${\left(s_0^{\prime }\right)}^2=1$ gives $X^{{\epsilon }_1}{Y}^{{\epsilon }_1^{\vee }}=q{Y}^{{\epsilon }_1^{\vee }}{X}^{{\epsilon }_1},$ which establishes the third relation of (Daff 3.6). $\square$

The group algebra of $\tilde{W}$ contains two Laurent polynomial rings This is analogous to the Weyl algebra generated by the two polynomial rings $$ℂ[x_1,...,x_n] \; and\; ℂ[\frac{\partial }{\partial x_1},...,\frac{\partial }{\partial x_n}] \; with\; [\frac{\partial }{\partial x_j},x_i]={\delta }_{ij}.$$ CHECK AGAIN?

Affine root systems

The notation for affine root systems is $$W_S=\left\{s_a \right| a\in S\}andS=(S_1,S_2),$$ where $$S_1=\{a\in S | \frac{1}{2}a\notin S\}and{S}_2=\{a\in S | 2a\notin S\}.$$

Define $$\begin{array}{rclrcl}{O}_1& =& \{\pm {\epsilon }_i+r | 1\le i\le n,r\in \mathbb{Z}\}& O_2& =& \{\pm 2{\epsilon }_i+2r | 1\le i\le n,r\in \mathbb{Z}\}\\ O_3& =& \{\pm {\epsilon }_i+\frac{1}{2}(2r+1) | 1\le i\le n,r\in \mathbb{Z}\}& O_4& =& \{\pm 2{\epsilon }_i+2r+1 | 1\le i\le n,r\in \mathbb{Z}\}\\ O_5^{-}& =& \{\pm ({\epsilon }_i-{\epsilon }_j)+r | 1\le i<j\le n,r\in \mathbb{Z}\}& O_5^{+}& =& \{\pm ({\epsilon }_i+{\epsilon }_j)+r | 1\le i<j\le n,r\in \mathbb{Z}\}\end{array}$$ and $O_5=O_5^{-}\bigsqcup O_5^{+}$ so that the affine root systems of classical type are missing Note that $${\epsilon }_n\in O_1, 2{\epsilon }_n\in O_2, {\epsilon }_1+\frac{1}{2}\delta \in O_3, 2{\epsilon }_1+\delta \in O_4,$$ and $${\epsilon }_i-{\epsilon }_{i+1}\in O_5, \; for\; i=1,...,n-1.$$

Where the left notation is the notation in [M03, §1.3] and the right notation is that of [BT]. The Weyl groups of these are $$W_S=\{\begin{array}{ll}{W}_{C_n},& for\; types\; C_n,C_n^{\vee },B{C}_n,(B{C}_n,C_n),(C_n^{\vee },B{C}_n),(C_n^{\vee },C_n) ,\\ W_{B_n},& for\; types\; B_n,B_n^{\vee },(B_n,B_n^{\vee }) ,\\ W_{D_n},& for\; type\; D_n\end{array}\phantom{\}}$$ where $$W_{C_n}=W_0⋉{𝔥}_{\mathbb{Z}}\supseteq W_{B_n}=W_0⋉{\stackrel{\_}{𝔥}}_{\mathbb{Z}}\supseteq W_{D_n}=\stackrel{\_}{W_0}⋉{\stackrel{\_}{𝔥}}_{\mathbb{Z}}$$ with $$W_0=G_{2,1,n},\stackrel{\_}{W_0}=G_{2,2,n}$$ and $${𝔥}_{\mathbb{Z}}=\sum _{i=1}^n\mathbb{Z}{\epsilon }_i,and{\stackrel{\_}{𝔥}}_{\mathbb{Z}}=\{{\lambda }_1{\epsilon }_1+\cdots +{\lambda }_n{\epsilon }_n | {\lambda }_1+\cdots +{\lambda }_n=0 \mathrm{mod} 2\},$$ SAY SOMETHING ABOUT $W_S$ orbits on $S.$ HOW DO THESE COMPARE TO KAC AND SAHI-ION???

When $n=2$ define $$\begin{array}{rclrcl}{O}_1& =& \{\pm {\epsilon }_i+r | 1\le i\le 2,r\in \mathbb{Z}\}& O_2& =& \{\pm 2{\epsilon }_i+2r | 1\le i\le 2,r\in \mathbb{Z}\}\\ O_3& =& \{\pm {\epsilon }_i+\frac{1}{2}(2r+1) | 1\le i\le 2,r\in \mathbb{Z}\}& O_4& =& \{\pm 2{\epsilon }_i+2r+1 | 1\le i\le 2,r\in \mathbb{Z}\}\\ O_5^{-}& =& \{\pm ({\epsilon }_1-{\epsilon }_2)+r | r\in \mathbb{Z}\}& O_5^{+}& =& \{\pm ({\epsilon }_1+{\epsilon }_2)+r | r\in \mathbb{Z}\}\end{array}$$ Then the "classical" affine root systems of rank 2 are

When $n=1$ define $$\begin{array}{rclrcl}{O}_1& =& \{\pm {\epsilon }_1+r | r\in \mathbb{Z}\}& O_2& =& \{\pm 2{\epsilon }_1+2r | r\in \mathbb{Z}\}\\ O_3& =& \{\pm {\epsilon }_1+\frac{1}{2}(2r+1) | r\in \mathbb{Z}\}& O_4& =& \{\pm 2{\epsilon }_1+2r+1 | r\in \mathbb{Z}\}\end{array}$$ Then the "classical" affine root systems of rank 1 are

Notes and References

Where are these from?

References

References?

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