July 11, 2016
For today’s lecture, we letV be a finite-dimensional vector space overR, with positive- definite inner product.
Definition 1. LetΦbe a nonempty finite set of nonzero vectors inV. We say thatΦis a root systemif
(R1) Φ∩Rα={α,−α}for allα∈Φ, (R2) sαΦ = Φfor allα∈Φ.
Lemma 2. LetGbe a finite group acting transitively on a setΩ. LetGαdenote the stabi- lizer ofαinG, that is,
Gα ={g ∈G|g.α=α}.
Then the following are equivalent:
(i) Gacts simply transitively onΩ, (ii) for everyα∈Ω,Gα={1}, (iii) for someα∈Ω,Gα ={1}, (iv) |G|=|Ω|.
LetΦbe a root system in V, and letW = W(Φ) = hsα | α ∈ Φi. Recall thatS(Φ) denotes the set of simple systems inΦ. Fix∆∈ S(Φ), and define
C ={λ∈V |(λ, α)>0 (∀α∈∆)}, D={λ∈V |(λ, α)≥0 (∀α ∈∆)}.
Notation 3. For a subsetU ofV, define
StabW(U) ={w∈W |wλ=λ(∀λ∈U)}.
Lemma 4. (i) Ifλ∈D, then
StabW({λ}) = hsα |α∈∆, sαλ =λi.
(ii) Ifλ, µ∈D,w∈W andwλ =µ, thenλ=µ.
(iii) Ifλ∈C, thenStabW({λ}) ={1}.
(iv) Ifλ∈V, then
StabW({λ}) = hsα |α∈Φ, sαλ=λi.
Theorem 5. For eachλ∈V,|W λ∩D|= 1.
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