$K$-spherical horospherical averages on the Nagao quotient: tree combinatorics and exact discrepancy

Let \[ F=\mathbb{F}_q(\!(t^{-1})\!),\qquad G=\mathrm{SL}_2(F),\qquad \Gamma=\mathrm{SL}_2(\mathbb{F}_q[t]),\qquad X=\Gamma\backslash G, \] and let $K=\mathrm{SL}_2(\mathcal{O})$, where $\mathcal{O}=\mathbb{F}_q[\![t^{-1}]\!]$.

We study right $K$-spherical averages along the upper unipotent subgroup, the horospherical subgroup associated with the standard cusp, on the Nagao lattice quotient.

The basic observation is that the $K$-spherical projection converts two natural dynamical families - expanding translates of compact unipotent orbits and cusp-adapted truncations of dense unipotent orbits - into the same rooted descendant problem on the Bruhat--Tits tree.

In the even bipartite sector the limiting height law is the explicit probability measure \[ \rho^{\mathrm{ev}}(0)=\frac{q-1}{q},\qquad \rho^{\mathrm{ev}}(2m)=(q^2-1)q^{-2m-1}\qquad (m\ge 1). \] We prove an exact discrepancy formula: in the backward state the error is a pure top-shell term minus a missing tail, while in the forward state the error is a first-turn weighted sum of backward errors.

These formulas give quantitative $K$-spherical equidistribution for expanding translates of compact $U$-orbits and for dense-orbit truncations.

For compactly supported $K$-spherical observables in the expanding translates of compact orbits, the discrepancy is eventually exactly zero.

In the dense case the rate is controlled by the continued-fraction expansion of the boundary point attached to the orbit.