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.

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.

Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $U\in P$ a nilpotent element. The parabolic springer fibre associated to $(P,U)$ can be defined by $$ \mathcal{P}_U:=\{gP\in G/P \, | \, g^{-1}Ug\in P\}\subseteq G/P. $$ It is known that these varieties admit affine pavings; see for instance, this reference. It follows that $|\mathcal{P}_U(\mathbb{F}_q)|$ 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 here. Note that the usual Springer fibres are pure; thus, the polynomial counting the number of points is the same as the Poincare polynomial.

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.

Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $U\in P$ a nilpotent element. The parabolic springer fibre associated to $(P,U)$ can be defined by $$ \mathcal{P}_U:=\{gP\in G/P \, | \, g^{-1}Ug\in P\}\subseteq G/P. $$ It is known that these varieties admit affine pavings; see for instance, this reference. It follows that $|\mathcal{P}_U(\mathbb{F}_q)|$ 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 $f_{\lambda,\nu}$?

Note that these varieties are known to be pure because they admit an affine stratification. Thus, equivalently, we are asking for their Betti numbers. For usual Springer fibres (i.e. when $P=B$) an answer is provided here.

Let $P$ be a parabolic subgroup of $\mathrm{GL}_n$ and $U\in P$ a nilpotent element. The parabolic springer fibre associated to $(P,U)$ can be defined by $$ \mathcal{P}_U:=\{gP\in G/P \, | \, g^{-1}Ug\in P\}\subseteq G/P. $$ It is known that these varieties admit affine pavings; see for instance, this reference. It follows that $|\mathcal{P}_U(\mathbb{F}_q)|$ 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 here. Note that the usual Springer fibres are pure; thus, the polynomial counting the number of points is the same as the Poincare polynomial.