Spectrum of weighted adjacency operator on a non-uniform arithmetic quotient of $PG{L}_3$

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 ${\tilde{A}}_2$-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 $PG{L}_3$ 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.

21 pages, 3 figures, Any comments welcome! v2: Minor Correction