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Asymptotic Schur orthogonality for lattices
Abstract
Let π:G→ 𝒰(L2(G/P,ν)) be the boundary representation of a non-compact connected semisimple Lie group G with finite center on its Furstenberg-Poisson boundary (G/P,ν). Let Γ⊂ G be a lattice in G (uniform or not). We show that for any \emph{continuous} functions $φ,ψ,φ',ψ'$ on G/P, limn→∞1/|Γn|∑γ∈ Γn⟨π(γ)φ,ψ⟩⟨π(γ)φ',ψ'⟩̅/Ξ2(γ)=⟨ φ,φ'⟩⟨ψ,ψ'⟩̅, where Γn is a ball in Γ with radius n relative to a natural length function and with center the identity element of G, and Ξ is the restriction to Γ of the Harish-Chandra function of G. As a corollary, we deduce that when the real rank of G is one, then the analogous convergence holds for any $φ,ψ,φ',ψ'$ in L2(G/P,ν) if and only if the lattice Γ is uniform.
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BibTeX
@misc{bendikov2026asymptotic,
title = {{Asymptotic Schur orthogonality for lattices}},
author = {A. BENDIKOV and A. BOYER and CH. PITTET},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04330},
eprint = {2610.04330},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04330},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Asymptotic Schur orthogonality for lattices
AU - A. BENDIKOV
AU - A. BOYER
AU - CH. PITTET
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04330
UR - https://arxiv.org/abs/2610.04330
AB - Let π:G→ 𝒰(L^2(G/P,ν)) be the boundary representation of a non-compact connected semisimple Lie group G with finite center on its Furstenberg-Poisson boundary (G/P,ν). Let Γ⊂ G be a lattice in G (uniform or not). We show that for any \emph{continuous} functions $φ,ψ,φ',ψ'$ on G/P, lim_n→∞1/|Γ_n|∑_γ∈ Γ_n⟨π(γ)φ,ψ⟩⟨π(γ)φ',ψ'⟩̅/Ξ^2(γ)=⟨ φ,φ'⟩⟨ψ,ψ'⟩̅, where Γ_n is a ball in Γ with radius n relative to a natural length function and with center the identity element of G, and Ξ is the restriction to Γ of the Harish-Chandra function of G. As a corollary, we deduce that when the real rank of G is one, then the analogous convergence holds for any $φ,ψ,φ',ψ'$ in L^2(G/P,ν) if and only if the lattice Γ is uniform.
ER -