A Character Formula for Compact Elements (the Rank One Case)

In their 1997 paper, [SS2], Schneider and Stuhler gave a formula relating the value of an admissible character of a p-adic group at an elliptic element to the fixed point set of this element on the BruhatTits building. Here we give a similar formula which works for compact elements. Elliptic elements have finitely many fixed facets in the building but compact elements can have infinitely many. In order to deal with the compact case we truncate the building so that we only look at a bounded piece of it. We show that for compact elements the (finite) information contained in the truncated building is enough to recover all of the information about the character. This works since the fixed point set of a compact (non elliptic) element is periodic. The techniques used here are more geometric in nature than the algebraic ones used by Schneider and Stuhler. We recover part of their result as a special case.