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

Soonki Hong, Sanghoon Kwon

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 $(λ^+,λ^-)$ 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.

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

Abstract

We investigate the automorphic spectra of the natural weighted adjacency operator on the complex arising as a quotient of -type building. We prove that the set of non-trivial approximate eigenvalues of the weighted adjacency operators on the quotient induced from the colored adjacency operators on the building for contains the simultaneous spectrum of and another hypocycloid with three cusps. As a byproduct, we re-establish a proof of the fact that is not a Ramanujan complex, from a combinatorial aspect.

Paper Structure

This paper contains 6 sections, 14 theorems, 89 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Building and compact Ramanujan complex
  3. Non-uniform arithmetic quotient $PGL(3,\mathbb{F}_q[t])\backslash PGL(3,\mathbb{F}_q(\!(t^{-1})\!))$
  4. Colored adjacency operator with weights
  5. Eigenfunctions of the weighted adjacency operators
  6. Automorphic spectra of the weighted adjacency operators

Key Result

Theorem 1.1

Let $A_w^+$ be weighted adjacency operator on $L^2_w(\Gamma\backslash \mathcal{B}(G))$ induced from the colored adjacency operator $A^+$ on $\mathcal{B}(G)$. The spectrum of the operator $A_w^+$ contains $\Sigma_0\cup\Sigma_1\cup\Sigma_2$ where is a set of three distinct points, is a hypocycloid with three cusps $(q^{\frac{3}{2}}+q+q^{\frac{1}{2}})e^{\frac{2k\pi i}{3}}$ for $k=0,1,2$ and is a h

Figures (3)

  • Figure 1: Spectrum of $A_X$ on $PGL(2,\mathbb{F}_q[t])\backslash \mathcal{T}_{q+1}$
  • Figure 2: $\Sigma_0\cup\Sigma_1\cup\Sigma_2$ in complex plane
  • Figure 3: The fundamental domain for $\Gamma\backslash\mathcal{B}(G)$

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Definition 4.1
  • Proposition 4.2
  • ...and 19 more