Let $\mathfrak{g}$ be a finite dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$. Let $W$ be the associated Weyl group and let $\Phi$ be its root system. We write $\Phi^+$ for the set of positive roots in $\Phi$.

Fix a subset of simple roots $I$. We define $ {}^IW := \{w\in W: w<s_\alpha w \ \text{for all }\alpha\in I\}, $ where $<$ is the Bruhat ordering on $W$.

According to Section 3.2 of KOSTANT MODULES IN BLOCKS OF CATEGORY $\mathcal{O}_S$:

${}^IW$ is an interval (i.e., it has a least and a greatest element).

It is obviously that $e$ is the least element of ${}^IW$. How to show the fact that ${}^IW$ also has a greatest element? I have read Deodhar's paper mentioned in Section 3.2, but I cannot find a proof for that.

Maybe it is a silly question, in my opinion, the statement "${}^IW$ is an interval with least element $u$ and greatest element $v$" means the following: ${}^I W=[u,v]:=\{x\in{}^IW: u\le x \le v\}$, where $\le$ is the Bruhat ordering.

I understand a finite Coxeter group contains a unique longest word (which is a maximal element), but I think the $v$ in my interpretation is the maximum element: $x\le v$ for all $x\in {}^IW$, which may not equal to the maximal element $w_0$: $w_0\le x\implies x=w_0$. Is my interpretation correct or not?

Generalized quotients in Coxeter groups, Trans. Amer. Math. Soc.308(1988), pp. 1--37, since $W$ has a maximum element. (Okay, you'll have to substitute $w^{-1}$ for $w$ everywhere, since they are considering $W^I$ rather than $\left. ^I W \right.$) $\endgroup$ – darij grinberg Aug 2 '19 at 11:45Hecke algebras with unequal parameters, arXiv:math/0208154v2 and take $I = \varnothing$. $\endgroup$ – darij grinberg Aug 2 '19 at 12:27