Take the tour ×  Suppose $G$ is a connected semisimple linear
algebraic group with Lie algebra $\mathfrak{g}$ and $X$ is a
homogeneous $G$-space with isotropy subgroup $H$ (associated Lie
algebra $\mathfrak{h}$) that is normalised by a parabolic
subgroup $P$. Write $A=P/H$.  Working over $\mathbb{C}$, Borho
and Brylinski explain in Proposition 2.8 of "Differential
Operators on Homogeneous Spaces I" how to use the moment map
$T^\ast X\rightarrow \mathfrak{g}^\ast$ to induce a map from
$\pi\colon T^\ast X/A\rightarrow \mathfrak{g}^\ast$. We may also
understand this map as a map from the vector bundle
$G\times^P\mathfrak{h}^\perp$ to $\mathfrak{g}^\ast$. It is a
generalisation of the Springer resolution (which arises in case
$H=P$ is a Borel).  The same construction can be made over an
algebraically closed field of characterstic $p>0$. My question
is what is known about the dimensions of the fibres of $\pi$ in
this characteristic $p$ case? I am most interested in knowing
the largest possible dimension of a fibre