## Reflection groups

Let $𝔬$ be a commutative ring, $𝔽$ the field of fractions of $𝔬$ and $\stackrel{‾}{𝔽}$ the algebraic closure if $𝔽$.

A $𝔬$-reflection group is a pair $\left({𝔥}_{𝔬},{W}_{0}\right)$ such that

• ${𝔥}_{𝔬}$ is a free $𝔬$-module,
• ${W}_{0}$ is a finite subgroup of $\mathrm{GL}\left({𝔥}_{𝔬}\right)$ generated by reflections,
where, letting ${𝔥}_{\stackrel{‾}{𝔽}}=\stackrel{‾}{𝔽}{\otimes }_{𝔬}{𝔥}_{𝔬}$, a reflection is $s\in \mathrm{GL}\left({𝔥}_{\stackrel{‾}{𝔽}}\right)$ which is conjugate (in $\mathrm{GL}\left({𝔥}_{\stackrel{‾}{𝔽}}\right)$) to $\mathrm{diag}\left(\xi ,1,\dots ,1\right)$ with $\xi \ne 1$.

The group ${W}_{0}$ acts on ${𝔥}_{𝔬}^{*}={\mathrm{Hom}}_{𝔬}\left({𝔥}_{𝔬},𝔬\right)$ by $⟨wμ,λ∨⟩ = ⟨μ,w-1 λ∨⟩, where ⟨μ,λ∨⟩ = μ(λ∨),$ for $w\in {W}_{0}$, $\mu \in {𝔥}_{𝔬}^{*}$, and ${\lambda }^{\vee }\in {𝔥}_{𝔬}$. Let $R+={ s∈W0 | sis a reflection}.$

If $s$ is a reflection in ${W}_{0}$ let $\alpha \in {𝔥}_{\stackrel{‾}{𝔽}}^{*}$ and ${\alpha }^{\vee }\in {𝔥}_{\stackrel{‾}{𝔽}}$ be chosen such that

 $sμ= μ-⟨μ,α∨⟩α, and s-1λ∨ =λ∨- ⟨λ∨,α⟩ α∨,$ (rff)
so that the reflecting hyperplanes for $s$ are $𝔥s= {λ∨∈𝔥 | ⟨λ∨,α⟩ =0} and (𝔥*)s= {μ∈𝔥* | ⟨μ,α∨⟩ =0} .$ Since $sα=(1- ⟨α,α∨⟩) α=det𝔥*(s) α,$ it follows that if $\stackrel{\sim }{\alpha },{\stackrel{\sim }{\alpha }}^{\vee }$ is another choice in (rff) then there is a constant $\gamma \in \stackrel{‾}{𝔽}$ such that $\stackrel{\sim }{\alpha }=\gamma \alpha$ and ${\stackrel{\sim }{\alpha }}^{\vee }={\gamma }^{-1}{\alpha }^{\vee }$.

## 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.

## References

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