Let $W$ be the Weyl group of a semisimple algebraic group $G$. $I$ be the simple roots. $J\subset I$ generate a parabolic subgroup of $W$ denote by $W\_J$. $w^J$
is the shortest representative of $w$ in $W/W\_J$. Suppose we have $u^J\le v^J$, here $\le$ is the Bruhat order. Let $M=\{ x\in W\_J| u^{J}x\le v^{J}\}$. My question is whether $M=\{id \}$ always hold?



Parabolic subgroup of Weyl group
No. In $A\_2$ with $I=\{s\_1,s\_2\}$, take $J=\{s\_1\}$ and $u^J=id$ and $v^J=s\_1s\_2$. Then $M=W\_J$ because $s\_1\le s\_1s\_2$.


