Weyl length equals inversion number

A generic segment from a point of $C_{+}$ to a point of $w(C_{+})$ meets exactly the hyperplanes separating the two chambers, each once, and produces a chain $C_{+}=C_0,\dots ,C_m=w(C_{+})$. Inductively, if $C_{k-1}=u_{k-1}(C_{+})$, the crossed wall is $u_{k-1}(L_{{\alpha }_{i_k}})$ for some simple root ${\alpha }_{i_k}$, and the adjacent chamber is $C_k=u_{k-1}{s}_{i_k}(C_{+})$. Thus $C_m=u_m(C_{+})$ for $u_m=s_{i_1}\cdots s_{i_m}$; simple transitivity and $C_m=w(C_{+})$ give $u_m=w$, while [L3] gives $m=\mathrm{\ell }(w)$. Conversely, given any expression $w=s_{i_1}\cdots s_{i_l}$, the chain $C_k=s_{i_1}\cdots s_{i_k}(C_{+})$ crosses one wall at each step, so at most $l$ hyperplanes separate its endpoints and $\mathrm{\ell }(w)\le l$. Hence $\mathrm{\ell }(w)$ is the minimum number of simple reflections in an expression of $w$.

The simple transitivity of $W$ on chambers applied to the pair $(C_{+},C_{-})$ gives a unique element $w_0\in W$ with $w_0(C_{+})=C_{-}$.

For every positive root $\alpha$, $w_0(\alpha )$ is negative: if $x\in C_{+}$ then $w_{0}x\in C_{-}$ and $(w_{0}x,w_0\alpha )=(x,\alpha )>0$; a root $\beta$ satisfying $(y,\beta )>0$ for all $y\in C_{-}=-C_{+}$ is negative, because writing $y=-x$ with $x\in C_{+}$ gives $(x,-\beta )=(y,\beta )>0$ and hence $-\beta \in {\mathrm{\Phi }}^{+}$ by [L4] and the definition of $C_{+}$. Hence $w_0({\mathrm{\Phi }}^{+})\subseteq {\mathrm{\Phi }}^{-}$; since $w_0$ is a bijection of the finite set $\mathrm{\Phi }$ and $\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=\mathrm{\mid }{\mathrm{\Phi }}^{-}\mathrm{\mid }$, equality holds, $N(w_0)={\mathrm{\Phi }}^{+}$, and $\mathrm{\ell }(w_0)=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$.

Every $w\in W$ satisfies $N(w)\subseteq {\mathrm{\Phi }}^{+}$, hence $\mathrm{\ell }(w)\le \mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }=\mathrm{\ell }(w_0)$; so $w_0$ is a longest element. If $w$ is also longest then $\mathrm{\ell }(w)=\mathrm{\mid }{\mathrm{\Phi }}^{+}\mathrm{\mid }$ forces $N(w)={\mathrm{\Phi }}^{+}$, that is $w({\mathrm{\Phi }}^{+})={\mathrm{\Phi }}^{-}$; then for every $x\in C_{+}$ and every positive root $\alpha$ one has $(wx,\alpha )=(x,w^{-1}\alpha )<0$, because $w^{-1}\alpha \in {\mathrm{\Phi }}^{-}$ and $x\in C_{+}$ has negative inner product with every negative root; hence $wx\in C_{-}$, that is $w(C_{+})=C_{-}$; simple transitivity of $W$ on chambers then gives $w=w_0$, so the longest element is unique.