Distance in the Affine Buildings of SL_n and Sp_n

Abstract

For a local field $K$ and $n\ge 2$, 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 ${\Xi }_n$ ($n\ge 3$) 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 ${\Delta }_n$ ($n\ge 2$) close to a given special vertex in ${\Delta }_n$ (all the vertices in ${\Xi }_n$ are special) and establish analogues of our results for ${\Delta }_n$.

16 pages, 3 figures; minor corrections; accepted for publication in INTEGERS: The Electronic Journal of Combinatorial Number Theory