Fundamental Domain

Fundamental Domain

Fix a positive system $\Pi$, containing the simple system $\Delta$. Associated with each hyperbolic $H_{\alpha }$ are the open half-spaces $A_{\alpha }$ and $A_{\alpha }^{\prime }$, where $A_{\alpha }:=\{\lambda \in V\mid (\lambda ,\alpha )>0\}$ and $A_{\alpha }^{\prime }:=-A_{\alpha }$. Define $C:={\bigcap }_{\alpha \in \Delta }{A}_{\alpha }$. As an intersection of open convex sets, $C$ is itself open and convex. It is also a cone (closed under positive scalar multiples). Let $D$ be the closure $\bar{C}$, the intersection of closed half-spaces $H_{\alpha }\cup A_{\alpha }$. Thus $D=\{\lambda \in V\mid (\lambda ,\alpha )\ge 0\text{ for all }\alpha \in \Delta \}.$ $D$ is a closed convex cone. It’s in fact the fundamental domain for the action of $W$ on $V$, i.e. each $\lambda \in V$ is conjugate under $W$ to one and only one point in $D$.

Lemma

Each $\lambda \in V$ is $W$-conjugate to some $\mu \in D$. Moreover, $\mu -\lambda$ is a nonnegative $\mathbb{R}$-linear combination of $\Delta$.

Theorem

Fix $\Pi \supset \Delta$ (hence $D$), as above. (a). If $w\lambda =\mu$ for $\lambda ,\mu \in D$, then $\lambda =\mu$ and $w$ is a product of simple reflections fixing $\lambda$. In particular, if $\lambda \in C$, then the isotropy group of $\lambda$ is trivial. (b). $D$ is a fundamental domain for the action of $W$ on $V$. (c). If $\lambda \in V$, the isotropy group of $\lambda$ is generated by those reflections $s_{\alpha }$ ($\alpha \in \Phi$) which it contains. (d). If $U$ is any subset of $V$, then the subgroup of $W$ fixing $U$ pointwise is generated by those reflections $s_{\alpha }$ which it contains.

Proof (a) Do induction on $l(w)=n(w)$, in trivial cases $l(w)=0$, $w=e$, we’re done. For $n(w)>0$, $w$ send some simple root $\alpha \in \Delta$ to a negative root. $n(w{s}_{\alpha })=n(w)-1$, so with $\mu \in D$, $w\alpha <0$. $0\ge ⟨\mu ,w\alpha ⟩=⟨w^{-1}\mu ,\alpha ⟩=⟨\lambda ,\alpha ⟩\ge 0,$ so $(\lambda ,\alpha )=0$, which means $s_{\alpha }(\lambda )=\lambda$, so $w{s}_{\alpha }(\lambda )=w\lambda$, through induction we eventually get $\lambda =w\lambda =\mu$. (b) It follows from (a) and previous lemma. (c) Given $\lambda \in V$, through lemma, exist $w\in W$, $\mu \in D$ that $w\lambda =\mu$. By (a), the isotropy group $W^{\prime }$ of $\mu$ is generated by simple reflections it contains. So the isotropy group of $\lambda$ is $w^{-1}{W}^{\prime }w$, generated by $w^{-1}{s}_{\alpha }w=s_{w^{-1}\alpha }$, which means it’s also generated by those reflections it contains. (d)We consider $\{{\lambda }_1,\cdots ,{\lambda }_k\}$ as a basis for the span of $U$. $W$ fixes $U$ pointwise if and only if $W$ fixes $\text{span}\{{\lambda }_1,\cdots ,{\lambda }_k\}$ pointwise, so $W$ would just be the intersection of isotropy group of ${\lambda }_i$. Induction on $t$, the trivial case $t=1$ is just (c), so the isotropy group $W_1$ of ${\lambda }_1$ is just generated by the set of all reflections $s_{\alpha }$ it contains, $\alpha$ forming a subset ${\Phi }_1\subset \Phi$, $W_1$ stabilizes ${\Phi }_1$ . So consider the Coxeter system $(W_1,{\Phi }_1)$. By induction hypothesis, the subgroup $W_0$ of $W_1$ fixing $\text{span}\{{\lambda }_2,\cdots ,{\lambda }_k\}$ is generated by reflections $s_{\alpha }$, $\alpha \in {\Phi }_1$. So the group fixes $\text{span}\{{\lambda }_1,\cdots ,{\lambda }_k\}$ pointwise is just $W_0$