Let $G$ be a reductive group over an (algebraically closed ) field. To each parabolic subgroup $P \subseteq G$ and $x \in G$ we can consider two types of partial Springer fibers associated to it :
$$1)\mathcal{P}_{x,1}=\{\ g \in G/P \ | \ g^{-1}xg \in U_P \} $$ where $U_P$ is the unipotent radical of $P$ and
$$2)\mathcal{P}_{x,2}=\{\ g \in G/P \ | \ g^{-1}xg \in P_{uni} \} $$ where $P_{uni}$ is the unipotent subvariety of $P$.
I haven't really understood which one we should take as the correct generalization of the standard Springer fibers (for $P=B$ Borel subgroup). In the original article by Bohro-MacPherson I think they study the latter and they show its cohomology is given by the $W(L)$ invariant of the usual Springer fiber (where $L\subseteq P$ is a Levi subgroup). In the same way more or less we are able to compute its points over a finite field at least in the case $G=Gl_n$(as in this answer Number of points of parabolic Springer fibres).
What about the first type of fiber however? Is there a nice combinatorial expression for its number of points over a finite field? I'm really interested in the case $G=Gl_n$
