The affine Weyl group

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

Last update: 25 September 2012

The affine Weyl group

For $\lambda \in P,$ the translation in $\lambda$ is

$$\begin{array}{cc}\begin{array}{cccc}{t}_{\lambda }:& {𝔥}_{ℝ}^{*}& ⟶& {𝔥}_{ℝ}^{*}\\ & x& ⟼& x+\lambda \text{.}\end{array}& \text{(1.10)}\end{array}$$

The extended affine Weyl group $\stackrel{\sim }{W}$ is the group

$$\begin{array}{cc}\stackrel{\sim }{W}=\{w{t}_{\lambda }|w\in W,\lambda \in P\},& \text{(1.11)}\end{array}$$

with multiplication determined by the relations

$$\begin{array}{cc}{t}_{\lambda }{t}_{\mu }=t_{\lambda +\mu },\text{and}{t}_{w\lambda }w=w{t}_{\lambda },& \text{(1.12)}\end{array}$$

for $\lambda ,\mu \in P$ and $w\in W,$ and so $\stackrel{\sim }{W}$ is a semidirect product of $W$ and the group of translations $\{t_{\lambda }\mid \lambda \in P\}\text{.}$ It is the group of transformations of ${𝔥}_{ℝ}^{*}$ generated by the $s_{\alpha },$ $\alpha \in R^{+},$ and $t_{\lambda },$ $\lambda \in P\text{.}$ The affine Weyl group $W_{\text{aff}}$ is the subgroup of $\stackrel{\sim }{W}$ generated by the reflections

$$\begin{array}{cc}\begin{array}{c}{s}_{\alpha ,k}:{𝔥}_{ℝ}^{*}⟶{𝔥}_{ℝ}^{*}\text{in the hyperplanes}\\ H_{\alpha ,k}=\{x\in {𝔥}_{ℝ}^{*}\mid ⟨x,{\alpha }^{\vee }⟩=k\},\alpha \in R^{+},k\in \mathbb{Z}\text{.}\end{array}& \text{(1.13)}\end{array}$$

The reflections $s_{\alpha ,k}$ can be written as elements of $\stackrel{\sim }{W}$ via the formula

$$\begin{array}{cc}{s}_{\alpha ,k}=t_{k\alpha }{s}_{\alpha }=s_{\alpha }{t}_{-k\alpha }\text{.}& \text{(1.14)}\end{array}$$

The highest short root of $R$ is the unique element $\phi \in R^{+}$ such that the fundamental alcove

$$\begin{array}{cc}A=C\cap \{x\in {𝔥}_{ℝ}^{*}\mid ⟨x,{\phi }^{\vee }⟩<1\}& \text{(1.15)}\end{array}$$

is a fundamental region for the action of $W_{\text{aff}}$ on ${𝔥}_{ℝ}^{*}\text{.}$ The various fundamental chambers for the action of $W_{\text{aff}}$ on ${𝔥}_{ℝ}^{*}$ are ${\stackrel{\sim }{w}}^{-1}A,$ $\stackrel{\sim }{w}\in W_{\text{aff}}\text{.}$ The inversion set of $\stackrel{\sim }{w}\in \stackrel{\sim }{W}$ is

$$R\left(\stackrel{\sim }{w}\right)=\{H_{\alpha ,k}\mid H_{\alpha ,k}\text{is between}A\text{and}{\stackrel{\sim }{w}}^{-1}A\}\text{and}\ell \left(\stackrel{\sim }{w}\right)=\text{Card}\left(R\left(\stackrel{\sim }{w}\right)\right)$$

is the length of $\stackrel{\sim }{w}\text{.}$ If $w\in W$ and $\lambda \in P$ then

$$\begin{array}{cc}\ell \left(w{t}_{\lambda }\right)=\sum _{\alpha \in R^{+}}\mid ⟨\lambda ,{\alpha }^{\vee }⟩+\chi \left(w\alpha \right)\mid ,& \text{(1.16)}\end{array}$$

where, for a root $\beta \in R,$ set $\chi \left(\beta \right)=0,$ if $\beta \in R^{+},$ and $\chi \left(\beta \right)=1,$ if $\beta \in R^{-}\text{.}$

In type $C_2$ we have the picture.

Let

$$\begin{array}{cc}{H}_{{\alpha }_0}=H_{\phi ,1}\text{and}{s}_0=s_{\phi ,1}=t_{\varphi }{s}_{\varphi }=s_{\varphi }{t}_{-\varphi },& \text{(1.17)}\end{array}$$

and let $H_{{\alpha }_1},\dots ,H_{{\alpha }_n}$ and $s_1,\dots ,s_n$ be as in (1.9). Then the walls of $A$ are the hyperplanes $H_{{\alpha }_0},H_{{\alpha }_1},\dots ,H_{{\alpha }_n}$ and the group $W_{\text{aff}}$ has a presentation by generators $s_0,s_1,\dots ,s_n$ and relations

$$\begin{array}{cc}\begin{array}{cccc}{s}_i^2& =& 1& \text{for}0\le i\le n,\\ \underset{\underset{m_{ij}\text{factors}}{⏟}}{s_i{s}_j{s}_i\dots }& =& \underset{\underset{m_{ij}\text{factors}}{⏟}}{s_j{s}_i{s}_j\dots },& i\ne j,\end{array}& \text{(1.18)}\end{array}$$

where $\pi /m_{ij}$ is the angle between the hyperplanes $H_{{\alpha }_i}$ and $H_{{\alpha }_j}\text{.}$

Let $w_0$ be the longest element of $W$ and let $w_i$ be the longest element of the subgroup $W_{{\omega }_i}=\{w\in W\mid w{\omega }_i={\omega }_i\}\text{.}$ Let ${\phi }^{\vee }=c_1{\alpha }_1^{\vee }+\dots +c_n{\alpha }_n^{\vee }\text{.}$ Then let

$$\begin{array}{cc}\Omega =\{g\in \stackrel{\sim }{W}\mid \ell \left(g\right)=0\}=\left\{1\right\}\cup \{g_i\mid c_i=1\},\text{where}{g}_i=t_{{\omega }_i}{w}_i{w}_0,& \text{(1.19)}\end{array}$$

(see [Bou1981, VI § no. 3 Prop. 6]). Each element $g\in \Omega$ sends the alcove $A$ to itself and thus permutes the walls $H_{{\alpha }_0},H_{{\alpha }_1},\dots ,H_{{\alpha }_n}$ of $A\text{.}$ Denote the resulting permutation of $\{0,1,\dots ,n\}$ also by $g\text{.}$ Then

$$\begin{array}{cc}g{s}_i{g}^{-1}=s_{g\left(i\right)},\text{for}0\le i\le n,& \text{(1.20)}\end{array}$$

and the group $\stackrel{\sim }{W}$ is presented by the generators $s_0,s_1,\dots ,s_n$ and $g\in \Omega$ with the relations (1.18) and (1.20). In the setting of Example 1.1, $W_{{\omega }_1}=\{1,s_2\},$ $W_{{\omega }_2}=\{1,s_1\},$ $w_1=s_2,$ $w_2=s_1$ and $w_0=s_1{s}_2{s}_1{s}_2,$ and ${\phi }^{\vee }=2{\alpha }_1^{\vee }+{\alpha }_2^{\vee }$ so that $c_1=2,$ $c_2=1$ and $\Omega =\{1,g_2\}\cong \mathbb{Z}/2\mathbb{Z},$ where $g_2=t_{{\omega }_2}{s}_2{s}_1{s}_2\text{.}$

Acknowledgements

The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977 and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).

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