Let $G$ be a reductive p-adic group. Let $W$ be a weyl group. if $w$, and $w_o \in W$. I want to know in which case we have $w w_o w^{-1}= w_o$ ?

in case if $y(\theta)=\theta $ where $\theta$ is a subset of simple roots, and $w$ is the longest element in $W_\theta$, the subgroup of $W$ generated by $s_\alpha$ (simple reflection of $\alpha$), where $\alpha$ is in $\theta$.

thanks.