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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.

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  • $\begingroup$ Just to clarify, when you write the word "pure", are you asking about the weights of Frobenius on the etale cohomology or are you asking about dimension / heights of associated points ("pure" in the sense of EGA)? Given your second question, I assume you mean weights of Frobenius. $\endgroup$ Aug 31, 2022 at 11:38
  • $\begingroup$ I mean weights of the Frobenius. Thanks for the clarification. $\endgroup$
    – Dr. Evil
    Sep 1, 2022 at 2:05

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