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

TL;DR

This paper rederives the zeta function identity for complexes from $ ext{PGL}_3$ using representation theory, providing new criteria for Ramanujan complexes based on eigenvalues of certain operators.

Contribution

It introduces a representation-theoretic approach to analyze the zeta functions of complexes from $ ext{PGL}_3$, offering new eigenvalue criteria for Ramanujan complexes.

Findings

Reproves the zeta function identity via eigenvalue analysis

Provides equivalent eigenvalue criteria for Ramanujan complexes

Connects combinatorial and representation-theoretic methods

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.