The extended affine Weyl group and non-trivial alcove stabilizers

Suppose $\mathrm{\Phi }\mathrm{\ne }\mathrm{\varnothing }$, let $\mathrm{\Delta }=\{{\alpha }_1,\dots ,{\alpha }_r\}$ be a simple-root basis, and let ${\omega }_1^{\vee },\dots ,{\omega }_r^{\vee }$ be the dual basis for $B({\alpha }_i,{\omega }_j^{\vee })={\delta }_{ij}$. By [F10], every root is an integer combination of the simple roots, so the condition defining $P^{\vee }$ is equivalent to $B({\alpha }_i,\lambda )\in \mathbb{Z}$ for every $i$; hence $P^{\vee }={⨁}_i\mathbb{Z}{\omega }_i^{\vee }$. By [F1], the simple coroots form a real basis and generate $Q^{\vee }$. Their coordinate matrix in the ${\omega }_i^{\vee }$ basis has entries $C_{ij}=B({\alpha }_i,{\alpha }_j^{\vee })\in \mathbb{Z}$ and is invertible, since its columns are a real basis. Thus $Q^{\vee }$ corresponds to $C{\mathbb{Z}}^r\subseteq {\mathbb{Z}}^r$, and [F8] shows $P^{\vee }\mathrm{/}{Q}^{\vee }$ is finite of order $\mathrm{\mid }\det C\mathrm{\mid }$. It is abelian because both lattices are additive groups.

For the displayed $A_2$ set, the roots span ${\mathbb{R}}^2$, have squared norm $1$, and occupy the six directions that are multiples of $\pi \mathrm{/}3$. The mirror of a root reflection is perpendicular to its root, so if the root direction is $j\pi \mathrm{/}3$, reflection sends a root direction $k\pi \mathrm{/}3$ to $(2j+3-k)\pi \mathrm{/}3$ and preserves the six roots. The possible Cartan numbers are $2B(\beta ,\alpha )\in \{2,1,-1,-2\}$, and each root line contains only the two displayed signs. Thus [F9] verifies the reduced crystallographic root-system axioms. Choose ${\mathrm{\Phi }}^{+}=\{{\alpha }_1,{\alpha }_2,\theta \}$; these are exactly the roots positive on $\theta$, since $B(\theta ,{\alpha }_1)=B(\theta ,{\alpha }_2)=1\mathrm{/}2$ and $B(\theta ,\theta )=1$. Then ${\alpha }_1,{\alpha }_2$ are simple and $\theta$ is highest. The coroots are $2{\alpha }_1,2{\alpha }_2,2\theta$ by [F3]. The vectors $v_1=(1,1\mathrm{/}\sqrt{3})$ and $v_2=(0,2\mathrm{/}\sqrt{3})$ satisfy $B({\alpha }_i,v_j)={\delta }_{ij}$, so they are the dual coweight basis. In this basis the simple coroots have columns $(2,-1)$ and $(-1,2)$, giving the matrix $\left(\begin{array}{cc}2& -1\\ -1& 2\end{array}\right)$ of determinant $3$. By [F8], $P^{\vee }\mathrm{/}{Q}^{\vee }$ has order $3$. The class of $v_1$ is nonzero because solving this matrix equation for $(1,0)$ gives $(2\mathrm{/}3,1\mathrm{/}3)$, not an integer vector; also $3v_1=2{\alpha }_1^{\vee }+{\alpha }_2^{\vee }$. Hence $[v_1]$ generates the quotient.

The Weyl group preserves $P^{\vee }$, since $B(\alpha ,w\lambda )=B(w^{-1}\alpha ,\lambda )\in \mathbb{Z}$ for every root $\alpha$, and it preserves $Q^{\vee }$ because it permutes coroots. For a root reflection, $s_{\alpha }(\lambda )-\lambda =-B(\lambda ,{\alpha }^{\vee })\alpha =-B(\lambda ,\alpha ){\alpha }^{\vee }$. If $\lambda \in P^{\vee }$, the last coefficient is integral, so every root reflection, and hence all of $W$, acts trivially on $P^{\vee }\mathrm{/}{Q}^{\vee }$. By [F2], $\pi ((\lambda ,w)(\mu ,v))=\lambda +w\mu +Q^{\vee }=\lambda +\mu +Q^{\vee }=\pi (\lambda ,w)+\pi (\mu ,v)$, so $\pi :W_a^{\mathrm{e}\mathrm{x}\mathrm{t}}\to P^{\vee }\mathrm{/}{Q}^{\vee }$, $\pi (\lambda ,w)=\lambda +Q^{\vee }$, is a homomorphism; it is onto and has kernel $Q^{\vee }⋊W=W_a$ by [F4]. Thus $W_a⊴W_a^{\mathrm{e}\mathrm{x}\mathrm{t}}$ with the stated quotient. Translations by $P^{\vee }$ shift each wall level by the integer $B(\lambda ,\alpha )$, while $W$ permutes roots, so $W_a^{\mathrm{e}\mathrm{x}\mathrm{t}}$ preserves the wall arrangement and its alcoves.

Let $\mathrm{\Omega }={\mathrm{Stab}}_{W_a^{\mathrm{e}\mathrm{x}\mathrm{t}}}(A)$. If $g\in \mathrm{\Omega }$ and $\pi (g)=0$, then $g\in W_a$ and [F6] gives $g=1$; hence $\pi {\mathrm{\mid }}_{\mathrm{\Omega }}$ is injective. For any $\lambda \in P^{\vee }$, $t_{\lambda }(A)$ is an alcove by step 2.1. By [F7] choose $h\in W_a$ carrying $t_{\lambda }(A)$ to $A$. Then $h{t}_{\lambda }\in \mathrm{\Omega }$, and its quotient class is $\pi (h{t}_{\lambda })=\lambda +Q^{\vee }$, so $\pi {\mathrm{\mid }}_{\mathrm{\Omega }}$ is onto. This representative is unique by injectivity. The closure of $A$ is a full-dimensional product of simplices by [F5], so its vertices affinely span $E$. An affine isometry fixing all those vertices is the identity; therefore $\mathrm{\Omega }$, and hence the quotient via $\pi {\mathrm{\mid }}_{\mathrm{\Omega }}$, acts faithfully on the vertex set of $\bar{A}$. The action of $W_a^{\mathrm{e}\mathrm{x}\mathrm{t}}$ on alcoves is transitive because its subgroup $W_a$ is transitive by [F7]; if $C=q(A)$, then ${\mathrm{Stab}}_{W_a^{\mathrm{e}\mathrm{x}\mathrm{t}}}(C)=q\mathrm{\Omega }{q}^{-1}$.

By [F5], the fundamental $A_2$ alcove is the triangle interior cut out by $B(x,{\alpha }_1)>0$, $B(x,{\alpha }_2)>0$, and $B(x,\theta )<1$. The simple-root walls meet at $0$; solving $B(x,{\alpha }_2)=0$, $B(x,\theta )=1$ gives $v_1$, and solving $B(x,{\alpha }_1)=0$, $B(x,\theta )=1$ gives $v_2$. From the root-reflection formula in [F3], $s_{{\alpha }_1}(x_1,x_2)=(-x_1,x_2)$ and $s_{{\alpha }_2}(x_1,x_2)=(x_1\mathrm{/}2+\sqrt{3}{x}_2\mathrm{/}2,\sqrt{3}{x}_1\mathrm{/}2-x_2\mathrm{/}2)$. Thus $w=s_{{\alpha }_1}{s}_{{\alpha }_2}$ sends $v_1$ to $v_2-v_1$ and $v_2$ to $-v_1$. With $\eta =v_1\in P^{\vee }$, the element $g=t_{\eta }w$ therefore sends $0\mapsto v_1\mapsto v_2\mapsto 0$. It fixes the triangle's centroid and cyclically permutes its vertices and facet walls, so it is a 120-degree rotation. In particular $g\in \mathrm{\Omega }$ and $g\mathrm{\ne }1$. Since $\mathrm{\Omega }\cong P^{\vee }\mathrm{/}{Q}^{\vee }\cong C_3$ by steps 3.1 and 1.2, $\mathrm{\Omega }=⟨g⟩$ and has order $3$. Thus the extended action is not free on alcoves, and this $A_2$ extended group is strictly larger than the Coxeter group $W_a$ generated by its affine facet reflections.

If $\mathrm{\Phi }=\mathrm{\varnothing }$, spanning forces $E=0$, so $P^{\vee }=Q^{\vee }=0$ and both affine groups are trivial. In rank one, with $\mathrm{\Phi }=\{\pm \alpha \}$, the coweight generator $\omega =\alpha \mathrm{/}B(\alpha ,\alpha )$ satisfies $P^{\vee }=\mathbb{Z}\omega$ and ${\alpha }^{\vee }=2\omega$, so the quotient is $\mathbb{Z}\mathrm{/}2\mathbb{Z}$; $t_{\omega }{s}_{\alpha }$ swaps the two vertices of the fundamental interval and stabilizes it. For reducible $\mathrm{\Phi }$, the simple-root and simple-coroot bases, the lattices, their quotient, the affine groups, and the fundamental alcove split over the orthogonal components, so the preceding proof applies factorwise. The only choice in step 3.1 is the element whose existence follows from transitivity; injectivity makes it unique for each coset, and no axiom of choice is used.