Let $k$ be algebraically closed field of characteristic $p>0$ and $G$ a connected, simply connected, semisimple algebraic group over $k$ and suppose $p$ is a very good prime for $G$. Let $K$ be a closed subgroup of $G$ and suppose $K$ acts on the flag variety $X$ with finitely many orbits; denote this orbits $\{X_s| s \in S\}$.
Lastly, consider the moment map $\pi:T^*X \to \mathfrak{g}^{*}$ and let $Y=\pi^{-1} \mathfrak{k}^{\perp}$, where $\mathfrak{k}=$Lie($K)$. Is it anything that can be said above the irreducible components of $Y$? I know that when working over $\mathbb{C}$ the irreducible components of $Y$ are given by closures over the conormal bundles $\overline{T_{X_s}^*X}$ by a result of Borho and Brylinski.
I am looking for references where something like this is studied. I have read the original proof of Borho-Brylinski along with the book by Chriss and Ginzburg; I am interested specifically in characteristic $p > 0.$
Many thanks.