Jordan types of triangular matrices over a finite field

Let $λ$ be a partition of an integer $n$ and ${\mathbb F}_q$ be a finite field of order $q$. Let $P_λ(q)$ be the number of strictly upper triangular $n\times n$ matrices of the Jordan type $λ$. It is known that the polynomial $P_λ$ has a tendency to be divisible by high powers of $q$ and $Q=q-1$, and we put $P_λ(q)=q^{d(λ)}Q^{e(λ)}R_λ(q)$, where $R_λ(0)\neq0$ and $R_λ(1)\neq0$. In this article, we study the polynomials $P_λ(q)$ and $R_λ(q)$. Our main results: an explicit formula for $d(λ)$ (an explicit formula for $e(λ)$ is known, see Proposition 3.3 below), a recursive formula for $R_λ(q)$ (a similar formula for $P_λ(q)$ is known, see Proposition 3.1 below), the stabilization of $R_λ$ with respect to extending $λ$ by adding strings of 1's, and an explicit formula for the limit series $R_{\lambda1^\infty}$. Our studies are motivated by projected applications to the orbit method in the representation theory of nilpotent algebraic groups over finite fields.

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