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Quantum ergodicity in the Benjamini--Schramm limit for locally symmetric spaces

Farrell Brumley, Simon Marshall, Jasmin Matz, Carsten Peterson

Abstract

We prove that for almost all symmetric spaces $X$ and for any sequence of compact locally symmetric spaces $Y_n$ which is uniformly discrete, has a uniform spectral gap, and converges in the sense of Benjamini--Schramm to $X$, the joint eigenfunctions of all invariant differential operators on $Y_n$ delocalize on average when their spectral parameters are taken to lie in a fixed spectral window.

Quantum ergodicity in the Benjamini--Schramm limit for locally symmetric spaces

Abstract

We prove that for almost all symmetric spaces and for any sequence of compact locally symmetric spaces which is uniformly discrete, has a uniform spectral gap, and converges in the sense of Benjamini--Schramm to , the joint eigenfunctions of all invariant differential operators on delocalize on average when their spectral parameters are taken to lie in a fixed spectral window.

Paper Structure

This paper contains 57 sections, 48 theorems, 270 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Review of literature: the rank one case
  3. Our main result
  4. Relation with the preceding work of Brumley--Matz
  5. Acknowledgements
  6. Sketch of proof and main technical theorems
  7. Spectral and geometric reduction steps
  8. Bounds on intersection volumes (Sections \ref{['sec_int_vol']}-\ref{['sec:sph-fn']})
  9. Bounds for spherical functions (Section \ref{['sec:sph-fn']})
  10. Preliminaries
  11. Basic notation
  12. Subsets of simple roots and associated structures
  13. Measures and Jacobian factors
  14. Spherical functions, spherical transform, spherical inversion
  15. Bounds on the $c$-function and the spherical function
  16. ...and 42 more sections

Key Result

Theorem 1.1

Let $G$ be a product of non-compact connected centerless simple real Lie groups, $K$ be a maximal compact subgroup, and $X = G/K$ be the associated symmetric space. Let $\Gamma_n<G$ be a sequence of torsion free, cocompact, uniformly discrete, irreducible lattices. Suppose $Y_n = \Gamma_n \backslash There exists a finite $W$-stable set of hyperplanes $\{ P_i\}$ in $\mathfrak a^*$ such that, for an

Figures (2)

  • Figure 1: The barycentric subdivision when $r=3$.
  • Figure 2: The singular refinement of the barycentric subdivision when $r=3$ and $t>3$. The subregion $T_{\sigma,0}$ does not appear since it does not meet the affine simplex $\{x\in \mathbb R_{>0}^3: x_1+x_2+x_3=1\}$.

Theorems & Definitions (84)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Proposition 3.1
  • proof
  • Proposition 3.2: Theorem 5.9.4 of Gangolli_Varadarajan_88
  • Theorem 3.3
  • ...and 74 more