Counting ${\mathbb{F}}_q$-points of orbital varieties in ad-nilpotent ideals of type $A_n$

Abstract

Let ${\mathfrak{b}}_n({\mathbb{F}}_q)$ denote the Lie algebra of upper triangular $n\times n$ matrices over the finite field ${\mathbb{F}}_q$, and let ${\mathfrak{u}}_n({\mathbb{F}}_q)$ be the nilradical of ${\mathfrak{b}}_n$. For every ${\mathfrak{b}}_n({\mathbb{F}}_q)$-stable ideal $\mathfrak{a}$ of ${\mathfrak{u}}_n({\mathbb{F}}_q)$, and every partition $\mu$ of $n$, we prove two formulas for the number of elements of $\mathfrak{a}$ of Jordan type $\mu$: the first one is the Hall scalar product of a modified Hall-Littlewood function indexed by $\mu$ and a chromatic quasisymmetric function associated to $\mathfrak{a}$, and the second one is in terms of an explicit collection of standard tableaux. In the special case that $\mathfrak{a}$ is the nilradical ${\mathfrak{u}}_{\Lambda }({\mathbb{F}}_q)$ of the parabolic subalgebra associated to a composition $\Lambda$ of $n$, our first formula reduces to a result of Karp and Thomas: up to an explicit polynomial factor in $q$, the number of elements in ${\mathfrak{u}}_{\Lambda }({\mathbb{F}}_q)$ of Jordan type $\mu$ is equal to the coefficient of the monomial ${\mathsf{x}}^{\Lambda }$ in the specialization of the dual Macdonald symmetric function ${\mathrm{Q}}_{{\mu }^{\prime }}(\mathsf{x};q^{-1},t)$ at $t=0$. We give three applications: (1) a formula for the number of points of a nilpotent Hessenberg variety, (2) a formula for the number of $X\in {\mathfrak{u}}_{\Lambda }({\mathbb{F}}_q)$ that satisfy $X^2=0$, which in the special case $\Lambda =(1^n)$ is different from the Kirillov-Melnikov-Ekhad-Zeilberger formula, and (3) a formula for the number of double cosets ${\mathrm{U}}_1\backslash {\mathrm{GL}}_n({\mathbb{F}}_q)/{\mathrm{U}}_2$ where ${\mathrm{U}}_1$ and ${\mathrm{U}}_2$ are unipotent subgroups corresponding to two ${\mathfrak{b}}_n({\mathbb{F}}_q)$-stable ideals.

Major revisions in Sections 1 and 2. Theorem 1.6 in the current version is a new result