Joint effective equidistribution of partial lattices in positive characteristic

Abstract

Let $\nu$ be a place of a global function field $K$ over a finite field, with associated affine function ring $R_\nu$ and completion $K_\nu$, and let $1 \leq \mathfrak{m}<\textbf{d}$. The aim of this paper is to prove an effective triple joint equidistribution result for primitive partial $R_\nu$-lattices $\Lambda$ of rank $\mathfrak{m}$ in $K_\nu^{\;\textbf{d}}$ as their covolume tends to infinity: of their $K_\nu$-linear span $V_\Lambda$ in the rank-$\mathfrak{m}$ Grassmannian space of $K_\nu^{\;\textbf{d}}$; of their shape in the modular quotient by $\operatorname{PGL}_\mathfrak{m}(R_\nu)$ of the Bruhat-Tits buildings of $\operatorname{PGL}_\mathfrak{m}(K_\nu)$; and of the shape of $\Lambda^\perp$ in the similar quotient for $\operatorname{PGL}_{\textbf{d}-\mathfrak{m}}(K_\nu)$, where $\Lambda^\perp$ is the orthogonal partial $R_\nu$-lattice of rank $\textbf{d}-\mathfrak{m}$ in the dual space of $K_\nu^{\;\textbf{d}}$. The main tools are a new refined $\text{LU}$ decomposition by blocks of elements of $\operatorname{SL}_\textbf{d}(K_\nu)$, techniques of Gorodnik and Nevo for counting integral points in well-rounded families of subsets of algebraic groups, and computations of volumes of various homogeneous spaces associated with partial $R_\nu$-lattices.