## The affine Weyl group

Last update: 25 September 2012

## The affine Weyl group

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

$tλ: 𝔥ℝ* ⟶ 𝔥ℝ* x ⟼ x+λ. (1.10)$

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

$W∼= { wtλ|w∈W ,λ∈P } , (1.11)$

with multiplication determined by the relations

$tλtμ= tλ+μ,and twλw=w tλ, (1.12)$

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 $\left\{{t}_{\lambda }\phantom{\rule{0.2em}{0ex}}\mid \lambda \in P\right\}\text{.}$ 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

$sα,k: 𝔥ℝ*⟶𝔥ℝ* in the hyperplanes Hα,k= { x∈𝔥ℝ*∣ ⟨x,α∨⟩=k } ,α∈R+,k∈ℤ. (1.13)$

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

$sα,k= tkαsα= sαt-kα. (1.14)$

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

$A=C∩ { x∈𝔥ℝ*∣ ⟨x,φ∨⟩<1 } (1.15)$

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

$R(w∼)= { Hα,k∣ Hα,kis betweenA andw∼-1 A } and ℓ(w∼)=Card (R(w∼))$

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

$ℓ(wtλ)= ∑α∈R+ ∣ ⟨λ,α∨⟩+ χ(wα) ∣ , (1.16)$

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

$Hα0=Hφ,1 ands0= sφ,1=tϕ sϕ=sϕ t-ϕ, (1.17)$

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

$si2 = 1 for0≤i≤n, sisjsi… ⏟ mijfactors = sjsisj… ⏟ mijfactors , i≠j, (1.18)$

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 ${W}_{{\omega }_{i}}=\left\{w\in W\phantom{\rule{0.2em}{0ex}}\mid \phantom{\rule{0.2em}{0ex}}w{\omega }_{i}={\omega }_{i}\right\}\text{.}$ Let ${\phi }^{\vee }={c}_{1}{\alpha }_{1}^{\vee }+\dots +{c}_{n}{\alpha }_{n}^{\vee }\text{.}$ Then let

$Ω= { g∈W∼∣ℓ (g)=0 } ={1}∪ {gi∣ci=1}, wheregi=tωi wiw0, (1.19)$

(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 $\left\{0,1,\dots ,n\right\}$ also by $g\text{.}$ Then

$gsig-1= sg(i), for0≤i≤n, (1.20)$

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, ${W}_{{\omega }_{1}}=\left\{1,{s}_{2}\right\},$ ${W}_{{\omega }_{2}}=\left\{1,{s}_{1}\right\},$ ${w}_{1}={s}_{2},$ ${w}_{2}={s}_{1}$ 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 $\Omega =\left\{1,{g}_{2}\right\}\cong ℤ/2ℤ,$ 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).