The Zeta Functions of Complexes from $\PGL(3)$: a Representation-theoretic Approach

Abstract

The zeta function attached to a finite complex $X_{\Gamma }$ arising from the Bruhat-Tits building for $\PGL_3(F)$ was studied in \cite{KL}, where a closed form expression was obtained by a combinatorial argument. This identity can be rephrased using operators on vertices, edges, and directed chambers of $X_{\Gamma }$. In this paper we reprove the zeta identity from a different aspect by analyzing the eigenvalues of these operators using representation theory. As a byproduct, we obtain equivalent criteria for a Ramanujan complex in terms of the eigenvalues of the operators on vertices, edges, and directed chambers, respectively.

arXiv admin note: substantial text overlap with arXiv:0804.2305