ScalingStacks

Proof. Let $v\in W$. There is a unique decomposition $v=v^{\prime}v^{\prime\prime}$ where $\ell(v)=\ell(v^{\prime})+\ell(v^{\prime\prime})$, $v^{\prime\prime}\in W_{I}$ and $v^{\prime}\in W^{I}$.

We have $d(T_{v})=d(T_{v^{\prime}})T_{v^{\prime\prime}}+T_{v^{\prime}}d(T_{v^{\prime\prime}})$. If $u\in W$ and $u<v^{\prime}$, then $u{\not\in}w_{S}W_{I}$. It follows that

$$t_{S,I}(d(T_{v}))=t_{S,I}(T_{v^{\prime}}d(T_{v^{\prime\prime}}))=\delta_{v^{\prime},w^{I}}d(T_{v^{\prime\prime}})=d(t_{S,I}(T_{v})).$$

∎