Distance in the Affine Buildings of SL_n and Sp_n

Abstract

For a local field $K$ and $n \geq 2$, let $\Xi_n$ and $\Delta_n$ 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 $\Xi_n$ ($n \geq 3$) close (i.e., gallery distance 1) to a given vertex in $\Xi_n$ to the number of chambers in $\Xi_n$ containing the given vertex, proving a conjecture of Schwartz and Shemanske. We then consider the special vertices in $\Delta_n$ ($n \geq 2$) close to a given special vertex in $\Delta_n$ (all the vertices in $\Xi_n$ are special) and establish analogues of our results for $\Delta_n$.