Combinatorics of a reduced root datum

Since $⟨\alpha ,{\alpha }^{\vee }⟩=2$ by (rd1), every $\alpha \in \mathrm{\Phi }$ is nonzero, and $\mathrm{\Phi }$ is finite and spans $V_0$ by construction; each $s_{\alpha }$ preserves $V_0$, acts on $x\in V_0$ by $x\mapsto x-⟨x,{\alpha }^{\vee }⟩\alpha$, fixes the hyperplane $⟨,{\alpha }^{\vee }⟩=0$ and sends $\alpha$ to $-\alpha$. Restricting the coroots to $V_0$ gives functionals ${\bar{\alpha }}^{\vee }$ with $⟨\mathrm{\Phi },{\bar{\alpha }}^{\vee }⟩\subseteq \mathbb{Z}$ and $s_{\alpha }(\mathrm{\Phi })\subseteq \mathrm{\Phi }$, so $(V_0,\mathrm{\Phi })$ satisfies the root-system axioms with coroots ${\bar{\alpha }}^{\vee }$; reducedness of the root datum is exactly the reducedness of this root system. Since $W(R)$ is finite by (rd3) and acts on the finite-dimensional real vector space $V_0\otimes \mathbb{R}$, averaging an arbitrary inner product over $W(R)$ produces an invariant inner product. For an invariant inner product the relation $⟨x,{\bar{\alpha }}^{\vee }⟩=2(x,\alpha )\mathrm{/}(\alpha ,\alpha )$ holds for all $x$, and the right side is independent of the choice of invariant inner product; hence $s_{\alpha }{\mathrm{\mid }}_{V_0}$ is the orthogonal reflection with vector $\alpha$ of the Euclidean theory, and the image of $W(R)$ in $\mathrm{GL}(V_0)$ is precisely the Weyl group $W(\mathrm{\Phi })$ of the reduced crystallographic root system $(V_0,\mathrm{\Phi })$ in the sense of Weyl group.

The natural map $W(R)\to W(\mathrm{\Phi })$ is surjective because the reflections generating the latter are restrictions of the generators of $W(R)$. For injectivity let $w$ restrict to the identity on $V_0$. Each reflection acts as the identity on $V\mathrm{/}{V}_0$, where $V=X\otimes \mathbb{Q}$, since its difference from the identity has image in the root line. Consequently $D=w-1$ sends $V$ into $V_0$ and vanishes on $V_0$, so $D^2=0$. The finite group $W(R)$ gives $w^m=1$ for some positive integer $m$; the binomial identity $(1+D{)}^m=1+mD$ then forces $D=0$ over $\mathbb{Q}$. Thus $w=1$ and $W(R)\cong W(\mathrm{\Phi })$. This also transfers finiteness, faithful action on the roots, and generation by the simple reflections of any base.

By the identification $W(R)=W(\mathrm{\Phi })$ of step 2.1, [F2] transfers verbatim: the simple roots of a base $\mathrm{\Delta }$ form a basis, every root is an integral combination of $\mathrm{\Delta }$ with coefficients of one sign, the Weyl chambers are the connected components of the complement of the hyperplanes $⟨,{\alpha }^{\vee }⟩=0$, and $W(R)$ acts simply transitively on them. To compare with the chambers in the full character space, average an inner product on $V\otimes \mathbb{R}$ over the finite group $W(R)$. Its orthogonal decomposition is $(V_0\otimes \mathbb{R})\oplus V_0^{\perp }$. Each reflection fixes $V_0^{\perp }$ pointwise: its difference from the identity lies in $V_0$, while invariance of the form keeps that difference in $V_0^{\perp }$. Thus every coroot functional vanishes there, and the full chambers are products of the root-span chambers with $V_0^{\perp }$. The transferred action is therefore simply transitive also on the full chambers. Moreover the sign conditions defining positive systems and the indecomposability defining simple roots are the same in the two languages, so bases, positive systems and chambers correspond bijectively, and simple transitivity on chambers is equivalent to simple transitivity on bases.

By [F3] applied to $(V_0,\mathrm{\Phi })$ there is a unique longest element $w_0\in W(R)$ with $w_0({\mathrm{\Phi }}^{+})=-{\mathrm{\Phi }}^{+}$, and for every $w$ the inversion number $n(w)=\mathrm{\mid }{\mathrm{\Phi }}^{+}\cap w({\mathrm{\Phi }}^{-})\mathrm{\mid }$ equals the length in simple reflections. Further $n(w^{-1})=n(w)$, because length is the minimum word length and $w^{-1}=s_{i_m}\cdots s_{i_1}$ whenever $w=s_{i_1}\cdots s_{i_m}$ is minimal. Finally $n(w_{0}w)=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-n(w)$. For $\alpha \in {\mathrm{\Phi }}^{+}$ put $\beta =-w_0^{-1}\alpha \in {\mathrm{\Phi }}^{+}$; this is a bijection of ${\mathrm{\Phi }}^{+}$. The condition $\alpha \in w_{0}w({\mathrm{\Phi }}^{-})$ says $w^{-1}{w}_0^{-1}\alpha \in {\mathrm{\Phi }}^{-}$, equivalently $w^{-1}\beta \in {\mathrm{\Phi }}^{+}$. These are precisely the complement of the $n(w)$ positive roots whose image under $w^{-1}$ is negative. Therefore the required count is $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }-n(w)$. If the root datum is semisimple, then $\mathbb{Z}\mathrm{\Phi }$ has finite index in $X$, so $\mathbb{Q}\mathrm{\Phi }=X\otimes \mathbb{Q}=V$ and $\mathrm{\Phi }$ is a reduced root system in $V$ in the Euclidean sense by step 1.1; conversely if $\mathrm{\Phi }$ spans $X\otimes \mathbb{Q}$ then the finitely generated subgroup $\mathbb{Z}\mathrm{\Phi }$ has full rank, hence finite index, in $X$, so the root datum is semisimple.