Distance in the Affine Buildings of SL_n and Sp_n

Abstract

For a local field $K$ and $n \geq 2$, let $Ξ_n$ and $Δ_n$ denote the affine buildings naturally associated to the special linear and symplectic groups $\SL_n(K)$ and $\Sp_n(K)$, respectively. We relate the number of vertices in $Ξ_n$ ($n \geq 3$) close (i.e., gallery distance 1) to a given vertex in $Ξ_n$ to the number of chambers in $Ξ_n$ containing the given vertex, proving a conjecture of Schwartz and Shemanske. We then consider the special vertices in $Δ_n$ ($n \geq 2$) close to a given special vertex in $Δ_n$ (all the vertices in $Ξ_n$ are special) and establish analogues of our results for $Δ_n$.