Intersections of standard parabolics, the parabolic root subsystem, and global minimality of coset representatives

Intersections of standard parabolics. By [F1], $w\in W_I$ if and only if $S(w)\subseteq I$, and $w\in W_J$ if and only if $S(w)\subseteq J$; hence $w\in W_I\cap W_J$ if and only if $S(w)\subseteq I\cap J$, if and only if $w\in W_{I\cap J}$. This proves (1) for all subsets $I,J$, including $I\cap J=\mathrm{\varnothing }$, where $W_{\mathrm{\varnothing }}=\{1\}$ by [F1] and [F2].

Invariance of $V_I$ and one containment. Let $s\in I$. By [F4], $\rho (s)e_s=-e_s$ and $\rho (s)e_t=e_t-2B(e_t,e_s)e_s$ for $t\in I$, $t\mathrm{\ne }s$; every vector of $V_I$ is a linear combination of the $e_t$, $t\in I$, so $\rho (s)V_I\subseteq V_I$, and since $\rho (s)$ is invertible with $\rho (s{)}^{-1}=\rho (s)$, also $\rho (s)V_I=V_I$. Every $w\in W_I$ is a product of elements of $I$ and their inverses by [F10], and the latter are again elements of $I$ because $s^2=1$; multiplying the identities $\rho (s)V_I=V_I$ along such a product gives $\rho (w)V_I=V_I$. Consequently, for $w\in W_I$ and $s\in I$ the root $\rho (w)e_s$ lies in $\mathrm{\Phi }\cap V_I$, so ${\mathrm{\Phi }}_I\subseteq \mathrm{\Phi }\cap V_I$; for $I=\mathrm{\varnothing }$ this says ${\mathrm{\Phi }}_{\mathrm{\varnothing }}=\mathrm{\varnothing }=\mathrm{\Phi }\cap \{0\}$, since $0\mathrm{\notin }\mathrm{\Phi }$ by [F5] and $V_{\mathrm{\varnothing }}=\{0\}$ by [F11].

Setup of the converse containment. Let $\phi \in \mathrm{\Phi }\cap V_I$. By [F5] the root $\phi$ lies in ${\mathrm{\Phi }}_{+}$ or in ${\mathrm{\Phi }}_{-}$; replacing $\phi$ by $-\phi$, which is again a root in $\mathrm{\Phi }\cap V_I$ because $\mathrm{\Phi }$ and $V_I$ are stable under negation, we may assume $\phi \in {\mathrm{\Phi }}_{+}$, so by [F5] and [F11] we have $\phi ={\sum }_{r\in I}{\lambda }_r{e}_r$ with ${\lambda }_r>0$ for every $r\in \mathrm{supp}\phi$, and $\phi \mathrm{\ne }0$. Let $t:=t_{\phi }\in T$ be the reflection of the dictionary [F7], so that $\rho (t)=r_{\phi }$ and $B(\phi ,\phi )=1$ by [F4]. We prove $t\in W_I$ by strong induction on $n:=\mathrm{\ell }(t)\ge 1$. If $n=1$, then $t$ is a word of length one, hence $t=r$ for some $r\in S$; the dictionary gives $t_{e_r}=r$, so $t=t_{\phi }=t_{e_r}$ and the clause "$t_{\alpha }=t_{\beta }$ if and only if $\alpha =\pm \beta$" of [F7] yields $\phi =\pm e_r$; since $\phi$ and $e_r$ both lie in ${\mathrm{\Phi }}_{+}$ while ${\mathrm{\Phi }}_{+}$ and ${\mathrm{\Phi }}_{-}=-{\mathrm{\Phi }}_{+}$ are disjoint by [F5], $\phi =e_r$, and then $r\in \mathrm{supp}\phi \subseteq I$ by [F11], so $t=r\in W_I$.

Additivity on the left coset. Let $d\in W^I$. By [F2] one has $\mathrm{\ell }(ds)>\mathrm{\ell }(d)$ for all $s\in I$, so $d$ is the minimal representative of the left coset $d{W}_I$ characterized in [F1], and therefore $\mathrm{\ell }(du)=\mathrm{\ell }(d)+\mathrm{\ell }(u)$ for all $u\in W_I$.

The induction step. Assume $n:=\mathrm{\ell }(t)\ge 2$ in the situation of step 1.3 and that the claim holds for all roots in ${\mathrm{\Phi }}_{+}\cap V_I$ whose reflection has length $<n$. Since $1=B(\phi ,\phi )={\sum }_{r\in I}{\lambda }_{r}B(\phi ,e_r)$ by [F3] and [F4], and ${\lambda }_r>0$ for $r\in \mathrm{supp}\phi$, there is $r\in \mathrm{supp}\phi \subseteq I$ with $B(\phi ,e_r)>0$; fix such an $r$. If $\mathrm{\mid }\mathrm{supp}\phi \mathrm{\mid }=1$, then $\phi ={\lambda }_r{e}_r$ with ${\lambda }_r>0$, and $1=B(\phi ,\phi )={\lambda }_r^{2}B(e_r,e_r)={\lambda }_r^2$ gives ${\lambda }_r=1$, that is $\phi =e_r$ and $t=t_{e_r}=r$; then $\mathrm{\ell }(t)=1$, against $n\ge 2$, so necessarily $\mathrm{\mid }\mathrm{supp}\phi \mathrm{\mid }\ge 2$. Put ${\phi }^{\mathrm{\prime }}:=\rho (t)e_r=r_{\phi }(e_r)=e_r-2B(\phi ,e_r)\phi$; its coefficient at each $r^{\mathrm{\prime }}\in \mathrm{supp}\phi \setminus \{r\}$ equals $-2B(\phi ,e_r){\lambda }_{r^{\mathrm{\prime }}}<0$, and this set is nonempty, so ${\phi }^{\mathrm{\prime }}\in {\mathrm{\Phi }}_{-}$ by [F5]. Put $\psi :=\rho (r)\phi =\phi -2B(\phi ,e_r)e_r\in \mathrm{\Phi }\cap V_I$; its coefficient at each $r^{\mathrm{\prime }}\in \mathrm{supp}\phi \setminus \{r\}$ is ${\lambda }_{r^{\mathrm{\prime }}}>0$, so $\psi \mathrm{\notin }{\mathrm{\Phi }}_{-}$ and hence $\psi \in {\mathrm{\Phi }}_{+}$ by [F5], while $\psi \in V_I$ because $\phi ,e_r\in V_I$. By the dictionary [F7], $t_{\psi }=rtr$. Since $\rho (t)e_r={\phi }^{\mathrm{\prime }}\in {\mathrm{\Phi }}_{-}$, the root-length criterion [F6] gives $\mathrm{\ell }(tr)<\mathrm{\ell }(t)$ and then $\mathrm{\ell }(tr)=\mathrm{\ell }(t)-1$ by [F8]; by [F9], $\mathrm{\ell }(rt)=\mathrm{\ell }(tr)=\mathrm{\ell }(t)-1$. Moreover $\rho (rt)e_r=\rho (r){\phi }^{\mathrm{\prime }}$ has at each $r^{\mathrm{\prime }}\in \mathrm{supp}\phi \setminus \{r\}$ the coefficient $-2B(\phi ,e_r){\lambda }_{r^{\mathrm{\prime }}}<0$ of ${\phi }^{\mathrm{\prime }}$, because $\rho (r)$ alters only the coefficient of $e_r$; hence $\rho (rt)e_r\in {\mathrm{\Phi }}_{-}$ by [F5], and the criterion [F6] with $w:=rt$ gives $\mathrm{\ell }(t_{\psi })=\mathrm{\ell }(rtr)<\mathrm{\ell }(rt)$, that is $\mathrm{\ell }(t_{\psi })=\mathrm{\ell }(t)-2$ by [F8]. Since $\psi \in {\mathrm{\Phi }}_{+}\cap V_I$ and $\mathrm{\ell }(t_{\psi })<n$, the induction hypothesis applies and yields $t_{\psi }\in W_I$; then $t=r{t}_{\psi }r\in W_I$ because $r\in I$.

The two descriptions of the quotients. Let $d\in W$. If $d\in W^I$, then by step 1.4 $\mathrm{\ell }(dv)=\mathrm{\ell }(d)+\mathrm{\ell }(v)\ge \mathrm{\ell }(d)$ for every $v\in W_I$, with equality if and only if $\mathrm{\ell }(v)=0$, that is $v=1$. Conversely, if $\mathrm{\ell }(d)\le \mathrm{\ell }(dv)$ for all $v\in W_I$, take $v=s\in I$: then $\mathrm{\ell }(ds)\ge \mathrm{\ell }(d)$, and $\mathrm{\ell }(ds)\mathrm{\ne }\mathrm{\ell }(d)$ by [F8], so $\mathrm{\ell }(ds)>\mathrm{\ell }(d)$ for all $s\in I$, that is $d\in W^I$ by [F2]; hence $W^I=\{d:\mathrm{\ell }(d)\le \mathrm{\ell }(dv)\text{ for all }v\in W_I\}$. Applying this to $d^{-1}$ and using $W^I=({}^{I}W{)}^{-1}$ from [F2] and the identities $(d^{-1}v{)}^{-1}=v^{-1}d$ and $\mathrm{\ell }(x^{-1})=\mathrm{\ell }(x)$ from [F9], we get $d\in {}^{I}W$ if and only if $\mathrm{\ell }(d)\le \mathrm{\ell }(v^{-1}d)$ for all $v\in W_I$; as $v$ runs over $W_I$ so does $v^{-1}$, so this is the second displayed description.

The converse containment. We prove by strong induction on $n$ that every $\phi \in {\mathrm{\Phi }}_{+}\cap V_I$ satisfies $t_{\phi }\in W_I$, the cases $n=1$ and $n\ge 2$ being steps 1.3 and 2.1; the induction is well founded because the length values are natural numbers. First let $\phi \in {\mathrm{\Phi }}_{+}\cap V_I$. Then $t_{\phi }\in W_I$, and its positive root is $\phi$. Strong exchange furnishes a representation with letters in $I$: choosing a reduced expression $t_{\phi }=s_1\cdots s_m$ of $t_{\phi }$ inside $W_I$ with all $s_i\in I$ by [F1] and applying strong exchange [F7] to $w:=t_{\phi }$ and the reflection $t_{\phi }$, whose square is $1$ and has length $0<m$, produces an index $i$ with $t_{\phi }=s_1\cdots s_{i-1}{s}_i{s}_{i-1}\cdots s_1$ and $\phi =\rho (s_1\cdots s_{i-1})e_{s_i}$ with $s_1\cdots s_{i-1}\in W_I$ and $s_i\in I$. Hence $\phi \in {\mathrm{\Phi }}_I$ for positive $\phi$. If instead $\phi \in {\mathrm{\Phi }}_{-}\cap V_I$, apply this argument to $-\phi =\rho (w)e_s$ with $w\in W_I$, $s\in I$; then $\phi =\rho (ws)e_s\in {\mathrm{\Phi }}_I$, since $\rho (s)e_s=-e_s$ and $ws\in W_I$ by [F4]. Therefore $\mathrm{\Phi }\cap V_I\subseteq {\mathrm{\Phi }}_I$.

Every coset has a unique minimum. Let $w\in W$ and let $d\in W^I$ be the unique element of $W^I\cap w{W}_I$, so that $w=dv$ with $v\in W_I$ by [F1] and $w{W}_I=d{W}_I$. Every $x\in d{W}_I$ has the form $x=du$ with $u\in W_I$, and by step 1.4, $\mathrm{\ell }(x)=\mathrm{\ell }(d)+\mathrm{\ell }(u)\ge \mathrm{\ell }(d)$, with equality if and only if $\mathrm{\ell }(u)=0$, that is $u=1$ and $x=d$. If $d^{\mathrm{\prime }}\in W^I\cap d{W}_I$ is a second such element, write $d^{\mathrm{\prime }}=d{u}^{\mathrm{\prime }}$ and $d=d^{\mathrm{\prime }}u$ with $u,u^{\mathrm{\prime }}\in W_I$ (possible since $d^{\mathrm{\prime }}{W}_I=d{W}_I$); step 1.4 gives $\mathrm{\ell }(d^{\mathrm{\prime }})=\mathrm{\ell }(d)+\mathrm{\ell }(u^{\mathrm{\prime }})$ and $\mathrm{\ell }(d)=\mathrm{\ell }(d^{\mathrm{\prime }})+\mathrm{\ell }(u)$, so $\mathrm{\ell }(u)=\mathrm{\ell }(u^{\mathrm{\prime }})=0$, $u=u^{\mathrm{\prime }}=1$ and $d=d^{\mathrm{\prime }}$. Thus $d{W}_I=w{W}_I$ contains exactly one element of $W^I$, namely its unique element of minimum length, which proves the right-handed assertions of (3). The left-handed assertions follow by the same argument applied to the inverses, using the description of ${}^{I}W$ in step 2.2 and the identities $\mathrm{\ell }(x^{-1})=\mathrm{\ell }(x)$ of [F9].

Conclusion of (2). Combining steps 1.2 and 3.1 gives ${\mathrm{\Phi }}_I=\mathrm{\Phi }\cap V_I$, which by [F11] is the displayed set of roots that are real linear combinations of the $e_s$, $s\in I$. Since $\mathrm{\Phi }={\mathrm{\Phi }}_{+}\bigsqcup {\mathrm{\Phi }}_{-}$ by [F5], intersecting with $V_I$ gives ${\mathrm{\Phi }}_I={\mathrm{\Phi }}_I^{+}\bigsqcup {\mathrm{\Phi }}_I^{-}$ for ${\mathrm{\Phi }}_I^{\pm }={\mathrm{\Phi }}_I\cap {\mathrm{\Phi }}^{\pm }={\mathrm{\Phi }}^{\pm }\cap V_I$. For the reflection identification: if $\alpha \in {\mathrm{\Phi }}_I^{+}$, say $\alpha =\rho (w)e_s$ with $w\in W_I$, $s\in I$, then $t_{\alpha }=ws{w}^{-1}\in W_I$ by [F7]; conversely, if $t\in W_I\cap T$, then $t=t_{\alpha }$ for the unique $\alpha \in {\mathrm{\Phi }}_{+}$ by [F7], and choosing a reduced expression $t=s_1\cdots s_m$ inside $W_I$ with letters in $I$ by [F1] and applying strong exchange [F7] to $w:=t$ and the reflection $t$ exhibits $t=s_1\cdots s_{i-1}{s}_i{s}_{i-1}\cdots s_1$ and $\alpha =\rho (s_1\cdots s_{i-1})e_{s_i}$, a root in ${\mathrm{\Phi }}_I$ because $s_1\cdots s_{i-1}\in W_I$ and $s_i\in I$. Hence the reflections of $W_I$ are exactly the $t_{\alpha }$ with $\alpha \in {\mathrm{\Phi }}_I^{+}$. Together with steps 1.1 and 3.2 this proves (1), (2) and (3); no Axiom of Choice is used: each proof fixes finitely many witnesses, and no simultaneous selection from an arbitrary family is required.