Algebraic Geometry – Number of Points of Parabolic Springer Fibres

Question

Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $u\in P$ a unipotent element. The parabolic Springer fibre associated to $(P,u)$ can be defined by
$$
\mathcal{P}_u:=\{gP\in G/P \mathrel\vert g^{-1}u g\in P\}\subseteq G/P.
$$
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. It follows that $\lvert\mathcal{P}_U(\mathbb{F}_q)\rvert$ is a polynomial in $q$. This polynomial depends only on the partitions $\lambda$ and $\nu$ associated to $P$ and $U$. Thus, we can denote it by $f_{\lambda,\nu}$.

Question: Is there an explicit formula for $f_{\lambda,\nu}$?

For usual Springer fibres (i.e. when $P=B$) an answer is provided at Fresse – Betti numbers of Springer fibers in type A. Note that the usual Springer fibres are pure; thus, the polynomial counting the number of points is the same as the Poincaré polynomial.

Answer 1

The earliest reference I could find that works out the polynomial $f_{\lambda, \nu}(q)$ is the paper

R. Hotta, N. Shimomura "The Fixed Point Subvarieties of Unipotent Transformations on Generalized Flag Varieties and the Green Functions" Math. Ann. 241, 193-208 (1979)

where the authors provide a recursive description of a certain statistic on tableaux which gives the desired polynomial. I also like the calculation in

A. Lascoux, B. Leclerc, J.-Y. Thibon, "Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties" J. Math. Phys. 38(2), 1041-1068 (1997) (arxiv)

where things are stated a little more explicitly: Recall the Kostka polynomials $K_{\lambda,\mu}(q)$, and their modified anaolgues $\tilde{K}_{\lambda, \nu}(q)$ (the generating functions of the charge and cocharge statistic, respectively, see here). If you define the polynomial $$\tilde{Q}'_{\lambda}(X,q)=\sum_{\mu}\tilde{K}_{\lambda, \mu}(q)s_{\mu}(X)$$ then our $f_{\lambda,\nu}(q)$ is the coefficient of the monomial symmetric function $m_{\nu}$ when $\tilde{Q}'_{\lambda}$ is expressed in the monomial symmetric function basis. Thus, you can write $$f_{\lambda,\nu}(q)=\sum_{\mu}\tilde{K}_{\lambda, \mu}(q)K_{\mu,\nu}.$$ As a side note, calculating the cohomology of these parabolic Springer fibers (sometimes referred to as Spaltenstein varieties) used to be the only known proof for the nonnegativity of the coefficients of the Kostka polynomials. The combinatorial understanding of charge/cocharge came later. :)