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Quantum ergodicity on the Bruhat-Tits building for $\text{PGL}(3, F)$ in the Benjamini-Schramm limit

Carsten Peterson

Abstract

We study joint eigenfunctions of the spherical Hecke algebra acting on $L^2(Γ_n \backslash G / K)$ where $G = \text{PGL}(3, F)$ with $F$ a non-archimedean local field of arbitrary characteristic, $K = \text{PGL}(3, O)$ with $O$ the ring of integers of $F$, and $(Γ_n)$ is a sequence of torsion-free lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself. This result is a higher rank non-archimedean analogue of existing results for graphs and locally symmetric spaces. A recurring theme in the proof is the reduction of many computations to computing the sum of an exponential function over lattice points in a polytope; such expressions can subsequently be simplified using Brion's formula. Along the way of proving our main result we prove several other results which may be of independent interest including a "degenerate" version of Brion's formula which "interpolates" between the usual Brion's formula and the Ehrhart polynomial, an effective rate of convergence for the distribution of spectral parameters to the Plancherel measure under Benjamini-Schramm convergence, and a classification of relative positions of triples of points in buildings of type $\tilde{A}_2$.

Quantum ergodicity on the Bruhat-Tits building for $\text{PGL}(3, F)$ in the Benjamini-Schramm limit

Abstract

We study joint eigenfunctions of the spherical Hecke algebra acting on where with a non-archimedean local field of arbitrary characteristic, with the ring of integers of , and is a sequence of torsion-free lattices. We prove a form of equidistribution on average for eigenfunctions whose spectral parameters lie in the tempered spectrum when the associated sequence of quotients of the Bruhat-Tits building Benjamini-Schramm converges to the building itself. This result is a higher rank non-archimedean analogue of existing results for graphs and locally symmetric spaces. A recurring theme in the proof is the reduction of many computations to computing the sum of an exponential function over lattice points in a polytope; such expressions can subsequently be simplified using Brion's formula. Along the way of proving our main result we prove several other results which may be of independent interest including a "degenerate" version of Brion's formula which "interpolates" between the usual Brion's formula and the Ehrhart polynomial, an effective rate of convergence for the distribution of spectral parameters to the Plancherel measure under Benjamini-Schramm convergence, and a classification of relative positions of triples of points in buildings of type .
Paper Structure (74 sections, 63 theorems, 187 equations, 6 figures, 1 table)

This paper contains 74 sections, 63 theorems, 187 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Quantum ergodicity in the large eigenvalue limit
  3. Quantum ergodicity in the Benjamini-Schramm limit
  4. Quantum ergodicity in higher rank
  5. Main result: quantum ergodicity in the Benjamini-Schramm limit for the Bruhat-Tits building associated to $\textnormal{PGL}(3, F)$
  6. Preliminaries
  7. Representation theory
  8. Coxeter complexes
  9. The Bruhat-Tits building
  10. Parametrization of $\Omega^+_{\textnormal{temp}}$
  11. Coloring eigenfunctions
  12. Benjamini-Schramm convergence
  13. Outline of the proof of Theorem \ref{['main_thm']}
  14. Notation
  15. Ancillary propositions and lemmas
  16. ...and 59 more sections

Key Result

Theorem 1.1

Suppose $(M, g)$ is a closed Riemannian manifold with ergodic geodesic flow. Then for every smooth test function $a \in C^\infty(M)$, where $N(\lambda) = \#\{i : \lambda_i \leq \lambda \}$.

Figures (6)

  • Figure 1: Geometric realization of the polytopes $P$, $P^*$, and $H$.
  • Figure 2:
  • Figure 4:
  • Figure 6:
  • Figure 8:
  • ...and 1 more figures

Theorems & Definitions (115)

  • Theorem 1.1: Quantum ergodicity theorem snirelmanzelditchcolin_de_verdiere
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: Quantum ergodicity on large regular graphs nalini_le_masson; see also brooks_le_masson_lindenstraussnalininalini_sabri
  • Remark 1.5
  • Theorem 1.6: Brumley-Matz brumley_matz
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 105 more