Hyperplanes for Affine Coxeter Group

Given a hyperplane $H=H_{\alpha ,k}\in \mathcal{H}$, each alcove $A\in \mathcal{A}$ lies in one or the other of the half-spaces defined by $H$. We say that $H$ separates two alcoves $A$ and $A^{\prime }$ if these alcoves lie in different half-spaces relative to $H$. For example, $H_s$ separates $A_0$ and $s{A}_0$, for each $s\in S_a$.

Note that, for a fixed pair of alcoves, the number of $H\in \mathcal{H}$ which separate them is finite: indeed, a bounded set (such as the line segment joining a pair of points from the two alcoves) obviously meets only finitely many of the parallel hyperplanes $H_{\alpha ,k}$ for each fixed $\alpha$. This allows us to define an integer-valued function on ${\hat{W}}_a$ by letting $n(w)$ be the cardinality of the set

$\mathcal{L}(w):=\{H\in \mathcal{H}\mid H\text{ separates }{A}_0\text{ and }w{A}_0\}.$

In the following section we shall show that the restriction of $n$ to $W_a$ is nothing but the length function $\ell$. Of course, $n(1)=0=\ell (1)$. It is also easy to see that $n(s)=1$ if $s\in S_a$, which amounts to showing that $\mathcal{L}(s)=\{H_s\}$: the line segment joining $\lambda \in A_0$ to $s\lambda$ meets no $H\in \mathcal{H}$ other than $H_s$.

As a further comparison with the length function, observe that $n(w)=n(w^{-1})$: $H$ separates $A_0$ and $w{A}_0$ if and only if $w^{-1}H$ separates $w^{-1}{A}_0$ and $A_0$, so $w^{-1}\mathcal{L}(w)=\mathcal{L}(w^{-1})$.

Note that, if $\mathcal{L}(w)$ is nonempty, then it must contain at least one of the hyperplanes $H_s,s\in S_a$. Otherwise $w{A}_0\ne A_0$ lies in the same open half-space as $A_0$ relative to each $H_s$. But $A_0$ is precisely the intersection of these half-spaces, yielding the contradiction $A_0=w{A}_0$.

Proposition

Let $w\in {\hat{W}}_a$ and fix $s\in S_a$.

• (a) $H_s$ belongs to exactly one of the sets $\mathcal{L}(w^{-1})$, $\mathcal{L}(s{w}^{-1})$.
• (b) $s(\mathcal{L}(w^{-1})\setminus \{H_s\})=\mathcal{L}(s{w}^{-1})\setminus \{H_s\}$.
• (c) $n(ws)=n(w)-1$ if $H_s\in \mathcal{L}(w^{-1})$, and $n(ws)=n(w)+1$ otherwise.

Corollary

For any $w\in W_a$, we have $n(w)\le \ell (w)$.

In order to prove that $\ell =n$ on $W_a$, as well as to show that $W_a$ acts simply transitively on $\mathcal{A}$, we want to write down an explicit list of the hyperplanes separating $A_0$ and $w{A}_0$. The key step is contained in the following lemma.

Lemma

If $w\ne 1$ in $W_a$ has a reduced expression $w=s_1\cdots s_r$, with $s_i\in S_a$, then (setting $H_i:=H_{s_i}$) the hyperplanes $H_1,s_1{H}_2,s_1{s}_2{H}_3,\dots ,s_1\cdots s_{r-1}{H}_r$ are all distinct.

Theorem

(a) Let $w\ne 1$ in $W_a$ have a reduced expression $w=s_1\cdots s_r$. Then we have (setting $H_i:=H_{s_i}$) $\mathcal{L}(w)=\{H_1,s_1{H}_2,s_1{s}_2{H}_3,\dots ,s_1\cdots s_{r-1}{H}_r\}.$ Moreover, these $r$ hyperplanes are all distinct.

• (b) The function $n$ on $W_a$ coincides with the length function $\ell$.
• (c) The group $W_a$ acts simply transitively on $\mathcal{A}$.