July 25, 2016
For today’s lecture, we letV be a finite-dimensional vector space overR, with positive- definite inner product. LetΦbe a root system inV, and let W = W(Φ) = hsα |α ∈Φi.
Fix a simple system∆inΦ, and letΠbe the unique positive system containing∆.
Lemma 1. Ifα∈∆, thensα(Π\ {α}) = Π\ {α}.
Define
C ={λ∈V |(λ, α)>0 (∀α∈∆)}, D={λ∈V |(λ, α)≥0 (∀α ∈∆)}.
Forα∈Φ, we define
Hα ={λ∈V |(α, λ) = 0}, Hα+={λ∈V |(α, λ)>0}, Hα−={λ∈V |(α, λ)<0}, so thatV =Hα+∪Hα∪Hα−(disjoint). Then
C= \
α∈∆
Hα+, D= \
α∈∆
(Hα+∪Hα).
Lemma 2. Forw∈W andα∈Φ,
wHα =Hwα, (1)
wHα± =Hwα± . (2)
In particular,
sαHα±=Hα∓, (3)
[
α∈Φ
Hα =w [
α∈Φ
Hα. (4)
Definition 3. The members of the family
{wC|w∈W} are calledchambers.
Proposition 4. IfU is a subset ofV, then
StabW(U) = hsα |α∈Φ, sα ∈StabW(U)i.
Proposition 5. LetΦbe a root system inV. Then the subgroup W(Φ) =hsα |α∈Φi
ofO(V)is a finite reflection group. Moreover,W(Φ)is essential if and only ifΦspansV. Conversely, for every finite reflection groupW ⊂ O(V), there exists a root systemΦ⊂V such thatW =W(Φ).
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