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Schur's orthogonality relations for the free group

Guillaume Delord

TL;DR

This work extends Schur's classical orthogonality from compact groups to the boundary representation $\pi$ of a free group acting on its boundary. It constructs $\pi$ on $\mathcal{H}=L^2(\Omega,\nu)$ via the conformal Radon–Nikodym factor and computes the Harish-Chandra function $\Xi$, proving its sphericality with an explicit closed form. The authors establish $c$-temperedness of $\pi$ using Følner balls and Harish-Chandra-type estimates, enabling a Kazhdan–Yom Din–type argument to derive asymptotic Schur orthogonality for matrix coefficients along balls $B_n$, with a precise cubic normalization constant. The main result is the limit formula for the averaged product of matrix coefficients, yielding an exact proportionality to $\langle\psi_1,\psi_3\rangle\overline{\langle\psi_2,\psi_4\rangle}$ and, after normalization, to $\frac{3q(q+1)}{(q-1)^2}$ times the same scalar, highlighting a non-compact analogue of Schur's relations in free-group boundary harmonic analysis.

Abstract

Explicit convergence of suitably normalized integrals on balls where the integrand is the product of coefficients of the quasi-regular representation of the finitely generated free group.

Schur's orthogonality relations for the free group

TL;DR

This work extends Schur's classical orthogonality from compact groups to the boundary representation of a free group acting on its boundary. It constructs on via the conformal Radon–Nikodym factor and computes the Harish-Chandra function , proving its sphericality with an explicit closed form. The authors establish -temperedness of using Følner balls and Harish-Chandra-type estimates, enabling a Kazhdan–Yom Din–type argument to derive asymptotic Schur orthogonality for matrix coefficients along balls , with a precise cubic normalization constant. The main result is the limit formula for the averaged product of matrix coefficients, yielding an exact proportionality to and, after normalization, to times the same scalar, highlighting a non-compact analogue of Schur's relations in free-group boundary harmonic analysis.

Abstract

Explicit convergence of suitably normalized integrals on balls where the integrand is the product of coefficients of the quasi-regular representation of the finitely generated free group.
Paper Structure (8 sections, 6 theorems, 51 equations, 2 figures)

This paper contains 8 sections, 6 theorems, 51 equations, 2 figures.

Table of Contents

  1. Introduction.
  2. Settings and notations.
  3. Organisation of the paper.
  4. Computing the Harish-Chandra function.
  5. Partitions where the Busemann function is constant.
  6. Computation: The Harish-Chandra function is spherical.
  7. $\pi$ is c-tempered.
  8. Asymptotic Schur's orthogonality relations for $\pi$.

Key Result

Lemma 2.1

$\textit{Let } x \in S_n \textit{ such that } [x_0,x] = (x_0,x_1,\dots, x=x_n)$. One defines the following sets: Then $\{E_0(x),\dots, E_n(x)\}$ is a partition of $\Omega$ (see Figure Dessin2 for an example where $n=2$). Moreover, $\omega \mapsto\beta_{\omega}(x_0,x)$ is constantly equal to $2k - n$ on $E_k(x)$.

Figures (2)

  • Figure 1: Example for $N=2$ and $x_0=e$
  • Figure 2: Example with $[x_0,x]=(x_0,x_1,x_2)$ illustrating Lemma \ref{['partition of finite geodesic']} where the family $(E_0,E_1,E_2)$ is a partition of the boundary.

Theorems & Definitions (13)

  • Lemma 2.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • Remark 4.2
  • Lemma 4.3
  • ...and 3 more