ScalingStacks

Let $R={\mathbf{Z}}[\{a_{s},b_{s}\}_{s\in S}]$ where $a_{s}$ and $b_{s}$ are indeterminates with $a_{s}=a_{s^{\prime}}$ and $b_{s}=b_{s^{\prime}}$ if $s$ and $s^{\prime}$ are conjugate in $W$.

The Hecke algebra $H=H(W)$ of $(W,S)$ is the $R$-algebra generated by $\{T_{s}\}_{s\in S}$ with relations

Given a reduced decomposition $w=s_{i_{1}}\cdots s_{i_{l}}$, we put $T_{w}=T_{s_{i_{1}}}\cdots T_{s_{i_{l}}}$. This element is independent of the choice of the reduced decomposition of $w$. The set $\{T_{w}\}_{w\in W}$ is a basis of $H$.

Let $\iota:H\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}H^{\operatorname{opp}\nolimits}$ be the algebra automorphism defined by $T_{s}\mapsto T_{s}$ for $s\in S$.

Let $I$ be a subset of $S$. We denote by $W_{I}$ the subgroup of $W$ generated by $I$. The group $W_{I}$, together with $I$, is a Coxeter group and the length function on $W_{I}$ is the restriction of that on $W$ [Hu, §1.10].

We put $R_{I}={\mathbf{Z}}[\{a_{s,I},b_{s,I}\}_{s\in I}]$ where $a_{s,I}$ and $b_{s,I}$ are indeterminates with $a_{s,I}=a_{s^{\prime},I}$ and $b_{s,I}=b_{s^{\prime},I}$ if $s$ and $s^{\prime}$ are conjugate in $W_{I}$. There is a morphism of rings $R_{I}\to R,\ a_{s,I}\mapsto a_{s},\ b_{s,I}\mapsto b_{s}$.

We denote by $H_{I}=H_{I}(W)$ the $R$-subalgebra of $H$ generated by $\{T_{s}\}_{s\in I}$. There is an isomorphism of $R$-algebras $R\otimes_{R_{I}}H(W_{I})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}H_{I}(W),\ T_{w}\mapsto T_{w}$.

We assume for the remainder of §3.1.2 that $W$ is finite. In this case, there is a unique element $w_{S}$ of $W$ with maximal length [Hu, §1.8] and we denote by $N$ its length. We have $w_{S}^{2}=1$ and $w_{S}Sw_{S}=S$. There is an automorphism of algebras

We denote by $w_{I}$ the longest element of $W_{I}$ and by $N_{I}$ its length. We denote by $W^{I}$ (resp. ${{}^{I}W}$) the set of elements $v\in W$ such that $v$ has minimal length in $vW_{I}$ (resp. $W_{I}v$). Note that $W^{I}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}W/W_{I},\ v\mapsto vW_{I}$ [Hu, Proposition 1.10].