Reflection groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 21 February 2012

Reflection groups

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

A $𝔬$-reflection group is a pair $({𝔥}_{𝔬},W_0)$ 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 ${𝔥}_{\overline{𝔽}}=\overline{𝔽}{\otimes }_{𝔬}{𝔥}_{𝔬}$, a reflection is $s\in \mathrm{GL}\left({𝔥}_{\overline{𝔽}}\right)$ which is conjugate (in $\mathrm{GL}\left({𝔥}_{\overline{𝔽}}\right)$) to $\mathrm{diag}(\xi ,1,\dots ,1)$ with $\xi \ne 1$.

The group $W_0$ acts on ${𝔥}_{𝔬}^{*}={\mathrm{Hom}}_{𝔬}({𝔥}_{𝔬},𝔬)$ by $$⟨w\mu ,{\lambda }^{\vee }⟩=⟨\mu ,w^{-1}{\lambda }^{\vee }⟩,\text{where}⟨\mu ,{\lambda }^{\vee }⟩=\mu \left({\lambda }^{\vee }\right),$$ for $w\in W_0$, $\mu \in {𝔥}_{𝔬}^{*}$, and ${\lambda }^{\vee }\in {𝔥}_{𝔬}$. Let $$R^{+}=\{s\in W_0|s\text{is a reflection}\}.$$

If $s$ is a reflection in $W_0$ let $\alpha \in {𝔥}_{\overline{𝔽}}^{*}$ and ${\alpha }^{\vee }\in {𝔥}_{\overline{𝔽}}$ be chosen such that

$$s\mu =\mu -⟨\mu ,{\alpha }^{\vee }⟩\alpha ,\text{and}{s}^{-1}{\lambda }^{\vee }={\lambda }^{\vee }-⟨{\lambda }^{\vee },\alpha ⟩{\alpha }^{\vee },$$

(rff)

so that the reflecting hyperplanes for $s$ are $${𝔥}^s=\{{\lambda }^{\vee }\in 𝔥|⟨{\lambda }^{\vee },\alpha ⟩=0\}\text{and}{\left({𝔥}^{*}\right)}^s=\{\mu \in {𝔥}^{*}|⟨\mu ,{\alpha }^{\vee }⟩=0\}.$$ Since $$s\alpha =(1-⟨\alpha ,{\alpha }^{\vee }⟩)\alpha ={\det }_{{𝔥}^{*}}\left(s\right)\alpha ,$$ it follows that if $\stackrel{\sim }{\alpha },{\stackrel{\sim }{\alpha }}^{\vee }$ is another choice in (rff) then there is a constant $\gamma \in \overline{𝔽}$ 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. ???

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