Parabolic subgroup

Definition

For $I\subseteq J\subseteq S$, let ${\mathcal{D}}_I^J\stackrel{\text{def}}{=}\{w\in W:I\subseteq D_R(w)\subseteq J\}$, $W^J\stackrel{\text{def}}{=}{\mathcal{D}}_{\varnothing }^{S\setminus J}$, ${\mathcal{D}}_I\stackrel{\text{def}}{=}{\mathcal{D}}_I^I$. Sets of the form ${\mathcal{D}}_I^J$ are called (right) descent classes. The special descent classes $W^J=\{w\in W:ws>w\text{ for all }s\in J\}$ are called quotients.

Proposition

(i) $(W_J,J)$ is a Coxeter group. (ii) ${\ell }_J(w)=\ell (w)$, for all $w\in W_J$. (iii) $W_I\cap W_J=W_{I\cap J}$. (iv) $⟨W_I\cup W_J⟩=W_{I\cup J}$. (v) $W_I=W_J⟹I=J$.

Lemma

An element $w$ belongs to $W^J$ if and only if no reduced expression for $w$ ends with a letter from $J$.

Proposition

Let $J\subseteq S$. Then, the following hold: (i) Every $w\in W$ has a unique factorization $w=w^J\cdot w_J$ such that $w^J\in W^J$ and $w_J\in W_J$. (ii) For this factorization, $\ell (w)=\ell (w^J)+\ell (w_J)$.