The longest element as the opposition of the chamber, and longest elements of finite parabolics

Since ${\mathrm{\Phi }}_{-}=-{\mathrm{\Phi }}_{+}$ [F3], the arrangement ${\bigcup }_{\alpha \in \mathrm{\Phi }}{H}_{\alpha }$ is invariant under the antipodal map $v\mapsto -v$ (as $H_{-\alpha }=H_{\alpha }$), and that map is a homeomorphism permuting the connected components of the complement; hence $-C^{\circ }$ is a connected component of $V\setminus {\bigcup }_{\alpha }{H}_{\alpha }$, and by the tiling [F2] there is a unique $w_0\in W$ with $-C^{\circ }=w_0{C}^{\circ }$; taking closures with [F2] gives $-C=w_{0}C$.

By [F1] the length is the least word length, so $\mathrm{\ell }(x)=0$ if and only if $x=1$; reversing a word gives $\mathrm{\ell }(w^{-1})=\mathrm{\ell }(w)$ for every $w$; and $\mathrm{\ell }(x)=1$ means that $x$ is represented by a one-letter word, that is, $x\in S$.

For $\alpha \in \mathrm{\Phi }$ one has $\alpha \in {\mathrm{\Phi }}_{+}$ exactly when $f(\alpha )>0$ for every $f\in C^{\circ }$, and $\alpha \in {\mathrm{\Phi }}_{-}$ exactly when $f(\alpha )<0$ for every $f\in C^{\circ }$ [F3]; as $f$ runs over $C^{\circ }$ the points $w_0\cdot f$ run over $-C^{\circ }$, and $(-g)(\alpha )=-g(\alpha )$, so $\alpha \in {\mathrm{\Phi }}_{+}$ is equivalent to $f(\rho (w_0{)}^{-1}\alpha )<0$ for every $f\in C^{\circ }$, that is, to $\rho (w_0{)}^{-1}\alpha \in {\mathrm{\Phi }}_{-}$; hence $\rho (w_0{)}^{-1}{\mathrm{\Phi }}_{+}={\mathrm{\Phi }}_{-}$, equivalently $\rho (w_0){\mathrm{\Phi }}_{+}={\mathrm{\Phi }}_{-}$, which is the equality $N(w_0)={\mathrm{\Phi }}_{+}$, and $\rho (w_0{)}^{-1}{\mathrm{\Phi }}_{+}={\mathrm{\Phi }}_{-}$ is $N(w_0^{-1})={\mathrm{\Phi }}_{+}$.

If $w^{\mathrm{\prime }}\in W$ satisfies $w^{\mathrm{\prime }}C=-C$, then $w^{\mathrm{\prime }}{C}^{\circ }=\mathrm{int}(w^{\mathrm{\prime }}C)=\mathrm{int}(-C)=-C^{\circ }=w_0{C}^{\circ }$, so $w^{\mathrm{\prime }}=w_0$ by the injectivity of $w\mapsto w{C}^{\circ }$ in [F2]; together with step 1.1 this proves uniqueness of the opposite chamber. If $v\in W$ satisfies $N(v)={\mathrm{\Phi }}_{+}$ or $N(v^{-1})={\mathrm{\Phi }}_{+}$, then $\rho (v{)}^{-1}{\mathrm{\Phi }}_{+}={\mathrm{\Phi }}_{-}$: in the first case $\rho (v){\mathrm{\Phi }}_{+}\subseteq {\mathrm{\Phi }}_{-}$ is equality because $\mathrm{\Phi }$ is finite and the two sign sets have equal cardinality, and negation gives $\rho (v){\mathrm{\Phi }}_{-}={\mathrm{\Phi }}_{+}$; in the second case this is the inversion-set definition. For $f\in C^{\circ }$ and $s\in S$, [F3] now gives $(v\cdot f)(e_s)=f(\rho (v{)}^{-1}{e}_s)<0$, so $v{C}^{\circ }\subseteq -C^{\circ }$. Both sets are connected components by [F2] and step 1.1, so they are equal and $v=w_0$ by [F2]. Conversely step 2.1 gives both full inversion sets for $w_0$, proving all equivalences in (i).

By [F4] one has $\mathrm{\ell }(w_0)=\mathrm{\mid }N(w_0)\mathrm{\mid }=\mathrm{\mid }{\mathrm{\Phi }}_{+}\mathrm{\mid }=\mathrm{\mid }T\mathrm{\mid }$, using step 2.1 and the bijection ${\mathrm{\Phi }}_{+}\to T$; and for every $w\in W$ the set $N(w_{0}w)=\{\alpha \in {\mathrm{\Phi }}_{+}:\rho (w_0)\rho (w)\alpha \in {\mathrm{\Phi }}_{-}\}$ equals $\{\alpha \in {\mathrm{\Phi }}_{+}:\rho (w)\alpha \in {\mathrm{\Phi }}_{+}\}={\mathrm{\Phi }}_{+}\setminus N(w)$, because $\rho (w_0)\beta \in {\mathrm{\Phi }}_{-}$ if and only if $\beta \in \rho (w_0{)}^{-1}{\mathrm{\Phi }}_{-}={\mathrm{\Phi }}_{+}$ by step 2.1; hence $\mathrm{\ell }(w_{0}w)=\mathrm{\mid }N(w_{0}w)\mathrm{\mid }=\mathrm{\mid }{\mathrm{\Phi }}_{+}\mathrm{\mid }-\mathrm{\mid }N(w)\mathrm{\mid }=\mathrm{\ell }(w_0)-\mathrm{\ell }(w)$ by [F4]. This proves (ii) and the first identity of (iii).

Taking $w=w_0$ in step 3.2 gives $\mathrm{\ell }(w_0^2)=\mathrm{\ell }(w_0)-\mathrm{\ell }(w_0)=0$, so $w_0^2=1$ by step 1.2; this is (iv).

By step 3.2, $0\le \mathrm{\ell }(w_{0}w)=\mathrm{\ell }(w_0)-\mathrm{\ell }(w)$ for every $w$, so $\mathrm{\ell }(w)\le \mathrm{\ell }(w_0)$ and $w_0$ has maximal length; if $\mathrm{\ell }(v)=\mathrm{\ell }(w_0)$ then $\mathrm{\ell }(w_0{v}^{-1})=\mathrm{\ell }(w_0)-\mathrm{\ell }(v^{-1})=\mathrm{\ell }(w_0)-\mathrm{\ell }(v)=0$ by steps 3.2 and 1.2, so $w_0{v}^{-1}=1$ by step 1.2 and $v=w_0$; hence $\mathrm{\ell }(w_0)=\max \{\mathrm{\ell }(w):w\in W\}$ and $w_0$ is the unique element of that length.

For every $w\in W$ the identity of step 3.2 applied to $w^{-1}$ gives $\mathrm{\ell }(w_0{w}^{-1})=\mathrm{\ell }(w_0)-\mathrm{\ell }(w^{-1})$, and inversion invariance together with $w_0^{-1}=w_0$ from step 4.1 gives $\mathrm{\ell }(w{w}_0)=\mathrm{\ell }(w_0)-\mathrm{\ell }(w)$ by step 1.2; this is the second identity of (iii).

For $s\in S$, step 5.1 with $w=s{w}_0$ gives $\mathrm{\ell }(s{w}_0{w}_0)=\mathrm{\ell }(w_0)-\mathrm{\ell }(s{w}_0)$, that is, $\mathrm{\ell }(s)=\mathrm{\ell }(w_0)-\mathrm{\ell }(s{w}_0)$ by step 4.1, so $\mathrm{\ell }(s{w}_0)=\mathrm{\ell }(w_0)-1$; then step 3.2 with $w=s{w}_0$ gives $\mathrm{\ell }(w_{0}s{w}_0)=\mathrm{\ell }(w_0)-\mathrm{\ell }(s{w}_0)=1$, so $w_{0}s{w}_0=:s^{\mathrm{\prime }}\in S$ by step 1.2. Conjugation by $w_0$ therefore maps $S$ into $S$, and since $w_0^2=1$ and $S$ is finite it maps $S$ bijectively onto $S$, that is, $w_{0}S{w}_0=w_{0}S{w}_0^{-1}=S$.

For $s\in S$ let $\sigma (s):=s^{\mathrm{\prime }}$, so that $w_{0}s{w}_0=\sigma (s)$ by step 6.1; applying the homomorphism $\rho$ and the conjugation formula [F5] gives $r_{\rho (w_0)e_s}=\rho (w_0)\rho (s)\rho (w_0{)}^{-1}=\rho (\sigma (s))=r_{e_{\sigma (s)}}$, that is, $t_{\rho (w_0)e_s}=t_{e_{\sigma (s)}}$ in the dictionary [F4]; since $t_{\alpha }=t_{\beta }$ for $\alpha ,\beta \in \mathrm{\Phi }$ only when $\alpha =\pm \beta$, one has $\rho (w_0)e_s=\pm e_{\sigma (s)}$, and the sign is negative because $\rho (w_0)e_s\in {\mathrm{\Phi }}_{-}$ by step 2.1 while $e_{\sigma (s)}\in {\mathrm{\Phi }}_{+}$ [F3]; hence $\rho (w_0)e_s=-e_{\sigma (s)}$ for every $s\in S$, and $\sigma$ is a permutation of $S$. This completes (v).

Let $I\subseteq S$ with $W_I$ finite. By [F6] the pair $(W_I,I)$ is a Coxeter system, its intrinsic length function is the restriction of $\mathrm{\ell }$, and $W_I\cap S=I$; moreover $I$ is finite. The canonical reflection representation of $(W_I,I)$ (on $V_I={\mathbb{R}}^I$ with the restricted Coxeter form $B_I=B{\mathrm{\mid }}_{V_I\times V_I}$) is the restriction of $\rho$ to $V_I$: each $s\in I$ has $\rho (s)e_t=e_t-2B(e_t,e_s)e_s\in V_I$ for $t\in I$ and $\rho (s)v=v$ for $v\in V_I^{\perp }$, so $\rho$ preserves $V_I$ and acts there by the reflection of $B_I$ with normal $e_s$, and both maps are homomorphisms on the presented group $(W_I,I)$ agreeing on generators, hence equal by the universal property [F7]. Therefore the argument of steps 1.1-7.1, which used only the tiling [F2], the root signs [F3], the inversion dictionary [F4], the conjugation formula [F5] and the length facts [F1] - all available verbatim for the Coxeter system $(W_I,I)$ with its intrinsic length - applies to $(W_I,I)$: there is a unique $w_0(I)\in W_I$ with $N_I(w_0(I))=({\mathrm{\Phi }}_I{)}_{+}$, it satisfies $w_0(I{)}^2=1$, it has maximal length in $W_I$ and is the unique element with $\mathrm{\ell }(w_0(I)w)=\mathrm{\ell }(w_0(I))-\mathrm{\ell }(w)$ and $\mathrm{\ell }(w{w}_0(I))=\mathrm{\ell }(w_0(I))-\mathrm{\ell }(w)$ for all $w\in W_I$, and $w_0(I)I{w}_0(I{)}^{-1}=I$. To verify the asserted uniqueness for the length-complement characterization, if $u\in W_I$ satisfies $\mathrm{\ell }(uw)=\mathrm{\ell }(u)-\mathrm{\ell }(w)$ for every $w\in W_I$, put $w=w_0(I)$ and $L:=\mathrm{\ell }(w_0(I))$. Then $0\le \mathrm{\ell }(u{w}_0(I))=\mathrm{\ell }(u)-L$, while maximality gives $\mathrm{\ell }(u)\le L$, so $\mathrm{\ell }(u)=L$ and uniqueness of the maximal-length element gives $u=w_0(I)$.

By the instance (ii) of the argument in step 8.1 one has $\mathrm{\ell }(w_0(I))=\mathrm{\mid }({\mathrm{\Phi }}_I{)}_{+}\mathrm{\mid }$, where $({\mathrm{\Phi }}_I{)}_{+}$ is the positive root system of the subsystem; by the intrinsic form of [F3] applied to $(W_I,I)$ (whose dual chamber is $\{g\in V_I^{\ast }:g(e_s)\ge 0\text{ for }s\in I\}$) a root $\alpha \in {\mathrm{\Phi }}_I\subseteq V_I$ is positive exactly when it lies in the subsystem's positive cone $V_I\cap V_{+}$, which is $V_I^{+}=\{{\sum }_{s\in I}{\lambda }_s{e}_s:{\lambda }_s\ge 0\}$; hence $({\mathrm{\Phi }}_I{)}_{+}={\mathrm{\Phi }}_I\cap V_I^{+}$ and $\mathrm{\ell }(w_0(I))=\mathrm{\mid }{\mathrm{\Phi }}_I\cap V_I^{+}\mathrm{\mid }$, as asserted in (2).