Spectrum of weighted adjacency operator on a non-uniform arithmetic quotient of $PGL_3$

TL;DR

This paper analyzes the automorphic spectra of weighted adjacency operators on a quotient of a building related to PGL(3), revealing spectral properties and confirming the non-Ramanujan nature of a specific complex.

Contribution

It characterizes the spectrum of weighted adjacency operators on a PGL(3) quotient, including the spectrum's shape and implications for Ramanujan properties.

Findings

Spectrum contains the simultaneous spectrum of adjacency operators.

Spectrum includes a hypocycloid with three cusps.

The quotient complex is not Ramanujan from a combinatorial perspective.

Abstract

We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a $PGL(3,\mathbb{F}_q[t])$ quotient of $\widetilde{A}_2$-type building. We prove that the set of non-trivial approximate eigenvalues $(\lambda^+,\lambda^-)$ of the weighted adjacency operators $A_w^\pm$ on the quotient induced from the colored adjacency operators $A^\pm$ on the building for $PGL_3$ contains the simultaneous spectrum of $A^\pm$ and another hypocycloid with three cusps. As a byproduct, we re-establish a proof of the fact that $PGL(3,\mathbb{F}_q[t])\backslash PGL(3,\mathbb{F}_q(\!(t^{-1})\!))/PGL(3,\mathbb{F}_q[\![t^{-1}]\!])$ is not a Ramanujan complex, from a combinatorial aspect.