Canonical affine geometry and wall parameters of a root datum

Average any inner product on $V=Y\otimes \mathbb{R}$ over the finite group $W_0$. Each reflection is then orthogonal, has normal line $\mathbb{R}{\alpha }^{\vee }$, and fixes $\ker \alpha$, so $⟨\alpha ,v⟩=2(v,{\alpha }^{\vee })\mathrm{/}({\alpha }^{\vee },{\alpha }^{\vee })$. Put $a_{\alpha }=2{\alpha }^{\vee }\mathrm{/}({\alpha }^{\vee },{\alpha }^{\vee })$. Its metric coroot is exactly ${\alpha }^{\vee }$, and $(v,a_{\alpha })=⟨\alpha ,v⟩$. Thus [F2] identifies the $a_{\alpha }$ as a reduced crystallographic root system on $E=\mathbb{R}{\mathrm{\Phi }}^{\vee }$, with the chosen positive system. The orthogonal complement $C=E^{\perp }$ is exactly the common root kernel and is fixed by $W_0$. Its presence adds a Cartesian factor $C$ to every alcove and changes neither the reflecting walls nor their combinatorics on $E$. Apply [F3]–[F4] on $E$ to obtain the asserted Coxeter system, simple transitivity and length for $Q^{\vee }⋊W_0$ on $V=E\oplus C$. The base alcove is equivalently $0<⟨\alpha ,v⟩<1$ for all positive $\alpha$: no integer wall cuts this region; every alcove in the positive chamber touching $0$ must satisfy these inequalities, and the supplied fundamental-alcove theorem identifies it with this region.

Every translation by $y\in Y$ sends $H_{\alpha ,j}$ to $H_{\alpha ,j+⟨\alpha ,y⟩}$ and every Weyl element permutes root directions, so $W_{\mathrm{e}}$ preserves the arrangement. Its conjugation preserves the group generated by all wall reflections, namely $W_{\mathrm{a}\mathrm{f}\mathrm{f}}$, hence that subgroup is normal. For $h\in W_{\mathrm{e}}$, simple transitivity gives a unique $w\in W_{\mathrm{a}\mathrm{f}\mathrm{f}}$ with $wA=hA$; then $w^{-1}h\in \mathrm{\Omega }$. The intersection is trivial by simple transitivity, proving the semidirect decomposition. Separating-wall length agrees with Coxeter length on the affine subgroup and vanishes exactly on $\mathrm{\Omega }$. If the roots are empty, $E=0$, $A=V$, $W_0=W_{\mathrm{a}\mathrm{f}\mathrm{f}}=1$, and $\mathrm{\Omega }=Y$, so every assertion still holds.

Every wall is a facet wall of some alcove by [F4]. Write that alcove as $wA$ with $w\in W_{\mathrm{a}\mathrm{f}\mathrm{f}}$; its reflection is $ws{w}^{-1}$ for a unique wall type $s$ of $A$. If two such descriptions give $ws{w}^{-1}=ut{u}^{-1}$, then $s$ and $t$ are conjugate in $W_{\mathrm{e}}$, so [F5] gives the same parameter. This defines $v(H)$ independently of the alcove used and makes it invariant under $W_{\mathrm{e}}$. Translation by ${\alpha }^{\vee }$ changes the integer level by $2$, so only parity can matter for fixed $\alpha$. If $\alpha \mathrm{\notin }2X$, perfect duality supplies $y\in Y$ with $⟨\alpha ,y⟩$ odd: otherwise evaluating on a lattice basis would put $\alpha \mathrm{/}2$ in $X$. Translation by that $y$ identifies odd and even walls, forcing equal parameters. If $\alpha \in 2X$, all pairings are even, and the formulas asserted are exactly the two potentially distinct parity values. The argument uses finitely many basis vectors and a finite Weyl average, without a choice axiom.