Let $G$ be a simple algrbraic group （ of type BCDEFG ） over the complex number $\mathbb{C}$, let $P$ be a parabolic subgroup of $G$, suppose we have a resolution of singularities $\mu: T^*(G/P)\to \bar{O}$, where $T^*(G/P)$ is the cotangent bundle of partial flag variety $G/P$ and $\bar{O}$ be Zariski closure of the Richardson nilpotent orbit $O$ corresponding to $P$ (this is not always a resolution of singularities, but we here consider it is. See Fu, Baohua Symplectic resolutions for nilpotent orbits. Invent. Math. 151 (2003), no. 1, 167–186. Journal Article )
Let $x$ be a nilpotent element in $\bar{O}$, $\mathcal{P}_x:=\mu^{-1}(x)$ we call it $\mathbf{parabolic\ Springer\ fiber}$.
My question: Is the connect component of $\mathcal{P}_x$ always equal dimensional in the case we consider above? Or give a counterexample please. Thanks!
