Counting $\mathbb F_q$-points of orbital varieties in ad-nilpotent ideals of type $A_n$ (2503.15440v2)

Abstract: Let $\mathfrak b_n(\mathbb F_q)$ denote the Lie algebra of upper triangular $n \times n$ matrices over $\mathbb F_q$, and let $\mathfrak u_n(\mathbb F_q)$ be the subalgebra of strictly upper triangular matrices. For every $\mathfrak b_n(\mathbb F_q)$-stable ideal $\mathfrak a \subseteq \mathfrak u_n(\mathbb F_q)$ and partition $\mu$ of $n$, we give an explicit formula for the number of elements in $\mathfrak a$ of Jordan type $\mu$. Up to a power of $q$, the number of points is given by the Hall scalar product of a modified Hall-Littlewood function indexed by $\mu$ and a chromatic quasisymmetric function associated to $\mathfrak a$. In the special case $\mathfrak a = \mathfrak u_\Lambda(\mathbb F_q)$, the nilradical of the standard parabolic subalgebra of $\mathfrak{gl}n(\mathbb F_q)$ corresponding to a composition $\Lambda$ of $n$, our formula specializes to a result of Karp and Thomas: up to a polynomial in $q$, the number of elements in $\mathfrak u\Lambda(\mathbb F_q)$ of Jordan type $\mu$ equals the coefficient of $\mathbf x^\Lambda$ in the specialization of the dual Macdonald symmetric function $\mathrm Q_{\mu'}(\mathbf x; q^{-1}, t)$ at $t = 0$. We give a new and shorter proof using a parabolic version of Borodin's division algorithm. We present four applications: (1) a formula for the number of points on a nilpotent Hessenberg variety; (2) a derivation of Kirillov's recurrence for counting nilpotent matrices of fixed Jordan type; (3) a formula for the number of $X \in \mathfrak u_\Lambda(\mathbb F_q)$ with $X^2 = 0$, yielding a new proof of the Kirillov-Melnikov-Ekhad-Zeilberger formula via two-row Macdonald polynomials; (4) a formula for the number of double cosets $\mathsf U_1 \backslash \mathsf{GL}_n(\mathbb F_q) / \mathsf U_2$, where $\mathsf U_1$ and $\mathsf U_2$ are unipotent subgroups from $\mathfrak b_n(\mathbb F_q)$-stable ideals.