The orbit of a dual fundamental functional: stabilizer, minimal coset length, Schreier distance, and the quotient formula

The functional $f_0$ vanishes exactly on $T$: $\{s\in S:f_0(e_s)=0\}=S\setminus \{s_0\}=T$, and all values $f_0(e_s)={\delta }_{s,s_0}$ are $\ge 0$, so $f_0\in C$. By [F3] the point stabilizer is ${\mathrm{Stab}}_W(f_0)=W_{S(f_0)}=W_T$.

For $u\in W_T$ one has $u\cdot f_0=f_0$ by step 1.1, so $w^{\mathrm{\prime }}\in w{W}_T$ (that is, $w^{\mathrm{\prime }}=wu$ with $u\in W_T$) implies $w^{\mathrm{\prime }}\cdot f_0=w\cdot (u\cdot f_0)=w\cdot f_0$: the orbit map $\mathrm{\Phi }$ is well defined. If $w\cdot f_0=w^{\mathrm{\prime }}\cdot f_0$, then applying $w^{-1}$ gives $w^{-1}{w}^{\mathrm{\prime }}\cdot f_0=f_0$, so $w^{-1}{w}^{\mathrm{\prime }}\in {\mathrm{Stab}}_W(f_0)=W_T$ and $w{W}_T=w^{\mathrm{\prime }}{W}_T$: $\mathrm{\Phi }$ is injective. It is surjective onto $W\cdot f_0$ by definition, and $\mathrm{\Phi }(w^{\mathrm{\prime }\mathrm{\prime }}w{W}_T)=w^{\mathrm{\prime }\mathrm{\prime }}\cdot \mathrm{\Phi }(w{W}_T)$ for all $w^{\mathrm{\prime }\mathrm{\prime }}$, so it is $W$-equivariant. Hence $\mathrm{\Phi }$ is a bijection. If $W\mathrm{/}{W}_T$ is finite, [F7] and this bijection give $\mathrm{\mid }W\cdot f_0\mathrm{\mid }=[W:W_T]$; in the infinite case the equality of sets remains the assertion, without finite-cardinality notation.

Fix $v\in W\cdot f_0$ and any $w$ with $w\cdot f_0=v$. For $x\in W$ one has $x\cdot f_0=w\cdot f_0$ if and only if $w^{-1}\cdot (x\cdot f_0)=f_0$, i.e. $(w^{-1}x)\cdot f_0=f_0$, i.e. $w^{-1}x\in {\mathrm{Stab}}_W(f_0)=W_T$, i.e. $x\in w{W}_T$. Thus $\{x\in W:x\cdot f_0=v\}=w{W}_T$.

The set $\{\mathrm{\ell }(x):x\in w{W}_T\}$ is a nonempty subset of $\mathbb{N}$, so it has a least element by [F5]; by step 2.2 that least element is $d(v)$. By [F2] the left coset $w{W}_T$ has a unique element $d_w$ of minimal length, characterized by $\mathrm{\ell }(d_{w}s)>\mathrm{\ell }(d_w)$ for all $s\in T$, and then $\mathrm{\ell }(d_{w}u)=\mathrm{\ell }(d_w)+\mathrm{\ell }(u)$ for all $u\in W_T$. Hence $d(v)=\mathrm{\ell }(d_w)$.

Let $f_0=v_0,v_1,\dots ,v_m=v$ be a walk in $\mathrm{\Gamma }$; an undirected edge may be traversed in reverse; the same generator still sends the preceding vertex to the next because its square is the identity by [F8], so there are $s_1,\dots ,s_m\in S$ with $v_i=s_i\cdot v_{i-1}$, so $v=s_m\cdots s_1\cdot f_0$ and $x:=s_m\cdots s_1$ satisfies $x\cdot f_0=v$ and $\mathrm{\ell }(x)\le m$. Therefore $d(v)\le \mathrm{\ell }(x)\le m$; taking the least such $m$ gives $d(v)\le {\mathrm{dist}}_{\mathrm{\Gamma }}(f_0,v)$.

Let $v\in W\cdot f_0$ and $s\in S$, and choose $x$ with $x\cdot f_0=v$ and $\mathrm{\ell }(x)=d(v)$ by step 3.1. Then $(sx)\cdot f_0=s\cdot v$, so $d(s\cdot v)\le \mathrm{\ell }(sx)\le \mathrm{\ell }(x)+1=d(v)+1$ by [F4]. Applying the same estimate to $s\cdot v$ in place of $v$ and using $s\cdot (s\cdot v)=(s^2)\cdot v=v$ gives $d(v)\le d(s\cdot v)+1$. Hence $\mathrm{\mid }d(s\cdot v)-d(v)\mathrm{\mid }\le 1$ for all $s,v$.

Conversely let $x=s_1\cdots s_m$ be a reduced expression with $x\cdot f_0=v$ and $m=d(v)$, which exists by step 3.1. The sequence $f_0,s_m\cdot f_0,s_{m-1}{s}_m\cdot f_0,\dots ,s_1\cdots s_m\cdot f_0=v$ is obtained by successively prepending letters. No consecutive vertices agree: if $s_i$ fixed the current suffix image $s_{i+1}\cdots s_m\cdot f_0$, deleting $s_i$ would still send $f_0$ to $v$ and give a representative of length at most $m-1$, contrary to $m=d(v)$. Thus every consecutive pair is an edge (self-loops are omitted), and ${\mathrm{dist}}_{\mathrm{\Gamma }}(f_0,v)\le m=d(v)$. With step 4.1 this gives ${\mathrm{dist}}_{\mathrm{\Gamma }}(f_0,v)=d(v)$; in particular $d(f_0)=0$ because $1\cdot f_0=f_0$ and $\mathrm{\ell }(1)=0$.

By [F2] every $w\in W$ has a unique factorization $w=d_{w}u$ with $u\in W_T$ and $d_w$ the minimal element of the left coset $w{W}_T$, and $\mathrm{\ell }(w)=\mathrm{\ell }(d_w)+\mathrm{\ell }(u)$. By steps 2.1, 2.2 and 3.1 the assignment $w\mapsto d_w\cdot f_0$ is a map of $W$ onto $W\cdot f_0$ whose fibers are exactly the cosets $w{W}_T$, and $d(d_w\cdot f_0)=\mathrm{\ell }(d_w)$. Summing $t^{\mathrm{\ell }(w)}=t^{d(d_w\cdot f_0)}{t}^{\mathrm{\ell }(u)}$ over the unique pairs $(d_w,u)$ therefore gives $P_W(t)={\sum }_{v\in W\cdot f_0}{\sum }_{u\in W_T}{t}^{d(v)+\mathrm{\ell }(u)}=\left({\sum }_{v\in W\cdot f_0}{t}^{d(v)}\right)P_{W_T}(t)$ in $\mathbb{Z}\mathrm{⟦}t\mathrm{⟧}$. Both factors are well-defined series by [F6]: for each $n$ the set $\{v:d(v)=n\}$ is contained in the image of the finite fiber $\{x:\mathrm{\ell }(x)=n\}$ under $x\mapsto x\cdot f_0$, hence finite.