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 $\mathrm Q_{\mu'}(\mathsf x;q^{-1},t)$ 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.