## The affine Weyl group action on ${𝔥}_{ℂ}^{*}$

Last update: 27 July 2013

## The affine Weyl group action on ${𝔥}_{ℂ}^{*}$

A reference for this section is [Kac1104219, §6.5].

Let $\stackrel{\circ }{R}$ be the set of roots of the finite dimensional Lie algebra $\stackrel{\circ }{𝔤}$ (see (1.10)). For $\alpha \in \stackrel{\circ }{R}$

$ν(hα)=α andν(hα∨) =α∨= 2α⟨α,α⟩, (1.21)$

where $\left[{e}_{\alpha },{f}_{\alpha }\right]={h}_{{\alpha }^{\vee }},$ and $\lambda \left({h}_{{\alpha }^{\vee }}\right)=⟨\lambda ,{\alpha }^{\vee }⟩\text{.}$

For $\alpha \in \stackrel{\circ }{R}$ and $k\in ℤ$ let

$h(-α+kδ)∨= -hα∨+k ⟨eα,fα⟩0 K=-hα∨+k12 ⟨hα∨,hα∨⟩0K$

and define

$s-α+kδ: 𝔥ℂ*⟶𝔥ℂ* bys-α+kδλ =λ-λ (h(-α+kδ)∨) (-α+kδ).$

Hence $\left[{e}_{\alpha },{f}_{\alpha }\right]={h}_{{\alpha }^{\vee }}$ and

$λ‾(hα∨) =⟨λ‾,α∨⟩$

relates the evaluation pairing and the form on ${𝔥}^{*}\text{.}$ If $\lambda =\alpha \delta +\stackrel{‾}{\lambda }+m{\Lambda }_{0}$ and ${s}_{\alpha }\stackrel{‾}{\lambda }=\stackrel{‾}{\lambda }-⟨\stackrel{‾}{\lambda },{\alpha }^{\vee }⟩$ then

$s-α+kδλ = λ-λ (h(-α+kδ)∨) (-α+kδ) = λ- (aδ+λ‾+mΛ0) (-hα∨+k12⟨hα∨,hα∨⟩0K) (-α+kδ) = λ-a·0- ⟨λ‾,-α∨⟩ (-α+kδ)-mk12 ⟨α∨,α∨⟩ (-α+kδ) = λ-⟨λ‾,α∨⟩α +mk⟨α,α∨⟩⟨α,α⟩ α+ ( ⟨λ‾,kα∨⟩- 12m ⟨kα∨,kα∨⟩ ) δ = λ-⟨λ‾,α∨⟩α +mkα∨+ ( ⟨λ‾,kα∨⟩- 12m ⟨kα∨,kα∨⟩ ) δ = ( a+ ⟨λ‾,kα∨⟩- 12m ⟨kα∨,kα∨⟩ ) δ+ λ‾- ⟨λ‾,α∨⟩ α+mkα∨+mΛ0 = ( a+ ⟨λ‾,kα∨⟩- 12m ⟨kα∨,kα∨⟩ ) δ+sα λ‾ α+mkα∨+mΛ0$

so that, in matrix form with respect to a basis $\delta ,{\stackrel{ˆ}{h}}_{1},\dots ,{\stackrel{ˆ}{h}}_{\ell },{\Lambda }_{0}$ of $𝔥,$ where ${\stackrel{ˆ}{h}}_{1},\dots ,{\stackrel{ˆ}{h}}_{\ell }$ is an orthonormal basis of $\stackrel{\circ }{𝔥},$

$s-α+kδ= ( 1 kα∨ -12 ⟨kα∨,kα∨⟩ ⋮ 0 ⋮ 000 0sα0 000 000 0kα∨0 000 0 … 0 … … 1 … )$

Then

$tkα∨= s-α+kδsα = ( 1 kα∨ -12 ⟨kα∨,kα∨⟩ ⋮ 0 ⋮ 000 0sα0 000 000 0kα∨0 000 0 … 0 … … 1 … ) ( 1 … 0 … 0 ⋮ 0 ⋮ 000 0sα0 000 ⋮ 0 ⋮ 0 … 0 … 1 ) = ( 1 -kα∨ -12 ⟨kα∨,kα∨⟩ ⋮ 0 ⋮ 000 010 000 000 0kα∨0 000 0 … 0 … … 1 … ) .$

Thus the group generated by the $\left\{{s}_{\alpha +k\delta } | \alpha \in \stackrel{\circ }{R},k\in ℤ\right\}$ is

$W=W0⋊Q∨= {wtμ∨ | w∈W0,μ∨∈Q∨} ,whereQ∨= ∑α∈R∘ ℤα∨.$

and ${W}_{0}$ is the group generated by the $\left\{{s}_{\alpha } | \alpha \in \stackrel{\circ }{R}\right\}\text{.}$