ScalingStacks

Proof. Consider $(j_{1},j_{2})\in L(\sigma)$ and let $s=s_{j_{1},j_{2}}$. Consider integers $i<j$ with $i-j{\not\in}n{\mathbf{Z}}$.

If $s(i)<s(j)$, then $(i,j)\in L(\sigma)$ if and only if $s(i,j)=(s(i),s(j))\in L(\sigma s)$.

Assume now $s(i)>s(j)$. We have three possibilities:

$\bullet\$ $i-j_{1}\in n{\mathbf{Z}}$, $j-j_{2}{\not\in}n{\mathbf{Z}}$: we have $(i,j)\in L(\sigma)$ if and only if $(i,j)\in L(\sigma s)$ or $\sigma(i)>\sigma(j)>\sigma s(i)$ (and then $(i,j){\not\in}L(\sigma s)$)

$\bullet\$ $i-j_{1}{\not\in}n{\mathbf{Z}}$, $j-j_{2}\in n{\mathbf{Z}}$: we have $(i,j)\in L(\sigma)$ if and only if $(i,j)\in L(\sigma s)$ or $\sigma s(j)>\sigma(i)>\sigma(j)$ (and then $(i,j){\not\in}L(\sigma s)$).

$\bullet\$ $i=j_{1}+nr$, $j=j_{2}+nr^{\prime}$ with $r,r^{\prime}\in{\mathbf{Z}}$: we have $(i,j)\in L(\sigma)$ if and only if $(i,j)\in L(\sigma s)$ or $\sigma(j_{1})-\sigma(j_{2})>n(r^{\prime}-r)>\sigma(j_{2})-\sigma(j_{1})$ (and then $(i,j){\not\in}L(\sigma s)$).

We deduce there is an injective map $a:L(\sigma s)\to L(\sigma)$ given by

$$a((i,j))=\begin{cases}(i,j)&\text{ if }s(i)>s(j)\\ s(i,j)&\text{ otherwise}\end{cases}$$

and

$$L(\sigma)=a(L(\sigma s))\sqcup\coprod_{|r|<\min\bigl(\frac{j_{2}-j_{1}}{n},\frac{\sigma(j_{1})-\sigma(j_{2})}{n}\bigr)}\bigl((j_{1}+nr,j_{2})+n{\mathbf{Z}}\bigr)\sqcup\\ \coprod_{\begin{subarray}{c}j_{1}<i<j_{2}\\ \sigma(j_{1})>\sigma(i)>\sigma(j_{2})\end{subarray}}\Bigl(\bigl((j_{1},i)+n{\mathbf{Z}}\bigl)\sqcup\bigl((i,j_{2})+n{\mathbf{Z}}\bigr)\Bigr).$$

Note that $a(L(\sigma s))\sqcup((j_{1},j_{2})+n{\mathbf{Z}})\subset L(\sigma)$.

Let us now prove the lemma. We have $\sigma=c^{d}w$ and $\sigma^{\prime}=c^{d^{\prime}}w^{\prime}\in\hat{{\mathfrak{S}}}_{n}$ for some $w,w^{\prime}\in W_{n}$. Assume $\sigma^{\prime}<\sigma$ and $\ell(\sigma^{\prime})=\ell(\sigma)-1$. We have $d=d^{\prime}$, $w^{\prime}<w$ and $\ell(w^{\prime})=\ell(w)-1$. It follows that there is a reduced decomposition $w=s_{a_{1}}\cdots s_{a_{l}}$ and $r\in\{1,\ldots,l\}$ such that $w^{\prime}=s_{a_{1}}\cdots s_{a_{r-1}}s_{a_{r+1}}\cdots s_{a_{l}}$. Let $j_{1}=s_{a_{l}}\cdots s_{a_{r+1}}(i_{r})$ and $j_{2}=s_{a_{l}}\cdots s_{a_{r+1}}(i_{r}+1)$. We have $(j_{1},j_{2})\in L(\sigma)$ and $\sigma^{\prime}=\sigma s_{j_{1},j_{2}}$ (Lemma 3.2.3).

The discussion above shows that $\{i\in{\mathbf{Z}}\ |\ j_{1}<i<j_{2},\ \sigma(j_{1})>\sigma(i)>\sigma(j_{2})\}=\emptyset$ and $\min\bigl(\frac{j_{2}-j_{1}}{n},\frac{\sigma(j_{1})-\sigma(j_{2})}{n}\bigr)<1$. The lemma follows. ∎