My question is the same as this post but for an arbitrary reductive $G$ instead of just $\mathrm{GL}_n$.
Let $G$ be a connected split reductive group over a finite field $k$. Let $P$ be a parabolic subgroup of $G$ and $u\in P$ a unipotent element. The parabolic Springer fibre associated to $(P,u)$ can be defined by $$ \mathcal{P}_u:=\{gP\in G/P \mathrel\vert g^{-1}u g\in P\}\subseteq G/P. $$
Question: Is this variety pure? Is it polynomial count? If so, is an explicit formula for the counting polynomial known?
For $G=\mathrm{GL}_n$ a complete answer is given in the above post. In classical types, it is known that these varieties admit affine pavings; see for instance, Fresse - Existence of affine pavings for varieties of partial flags associated to nilpotent elements.