Reflection groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last updates: 21 February 2012

Reflection groups

Let 𝔬 be a commutative ring, 𝔽 the field of fractions of 𝔬 and 𝔽 the algebraic closure if 𝔽.

A 𝔬-reflection group is a pair (𝔥𝔬,W0) such that

where, letting 𝔥𝔽 = 𝔽𝔬 𝔥𝔬, a reflection is sGL( 𝔥𝔽) which is conjugate (in GL( 𝔥𝔽)) to diag(ξ,1, ,1) with ξ1.

The group W0 acts on 𝔥𝔬*= Hom𝔬 (𝔥𝔬,𝔬) by wμ,λ = μ,w-1 λ, where μ,λ = μ(λ), for wW0, μ𝔥𝔬*, and λ𝔥𝔬. Let R+={ sW0 | sis a reflection}.

If s is a reflection in W0 let α 𝔥𝔽* and α𝔥𝔽 be chosen such that

sμ= μ-μ,αα, and s-1λ =λ- λ,α α, (rff)
so that the reflecting hyperplanes for s are 𝔥s= {λ𝔥 | λ,α =0} and (𝔥*)s= {μ𝔥* | μ,α =0} . Since sα=(1- α,α) α=det𝔥*(s) α, it follows that if α, α is another choice in (rff) then there is a constant γ 𝔽 such that α= γα and α =γ-1 α.

Notes and References

These notes are partly based on the definitions in [AG+] (Andersen, Grodal, etc). This paragraph is taken from Section 3 of stephen8.9.06.pdf.


[AG+] ?. Andersen, J. Grodal, ??????, Clasification of p-compact groups for p odd, Ann. Math. ???

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