Higher-rank local mixing for cusped positive representations
Abstract
Let Γ<PSL2(ℝ) be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let ρ:Γ→ G be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen--Margulis--Sullivan measures on ρ(Γ)backslash G/M, where M is the centralizer of a maximal split torus in a compatible maximal compact subgroup. For every u in the interior of the limit cone of ρ(Γ), correlations of smooth compactly supported functions along exp(tu), normalized by t(rankℝG−1)/2, admit a complete asymptotic expansion in integer powers of t−1. We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on L2(ρ(Γ)backslash G) and for orbital counting in Riemannian balls. More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of Γ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations. To prove these results, we develop an intrinsic cusp geometry on the limit set of ρ(Γ) in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.
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BibTeX
@misc{kim2026higherrank,
title = {{Higher-rank local mixing for cusped positive representations}},
author = {Dongryul M. Kim and Hee Oh and Wenyu Pan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12254},
eprint = {2610.12254},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12254},
}
RIS
TY - UNPB
TI - Higher-rank local mixing for cusped positive representations
AU - Dongryul M. Kim
AU - Hee Oh
AU - Wenyu Pan
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12254
UR - https://arxiv.org/abs/2610.12254
AB - Let Γ<PSL_2(ℝ) be a torsion-free, non-elementary, finitely generated Fuchsian group with parabolic elements, and let ρ:Γ→ G be a type-preserving, Zariski dense, positive representation into a connected split semisimple real algebraic group. We prove quantitative local mixing for Bowen--Margulis--Sullivan measures on ρ(Γ)backslash G/M, where M is the centralizer of a maximal split torus in a compatible maximal compact subgroup. For every u in the interior of the limit cone of ρ(Γ), correlations of smooth compactly supported functions along exp(tu), normalized by t^(rank_ℝG−1)/2, admit a complete asymptotic expansion in integer powers of t^−1. We also obtain a uniform Gaussian asymptotic for transverse displacements, with an integrable remainder. We deduce asymptotics for matrix coefficients of the quasi-regular representation on L^2(ρ(Γ)backslash G) and for orbital counting in Riemannian balls. More generally, these conclusions hold for Zariski dense, relatively Borel Anosov representations of Γ satisfying a peripheral root-balance condition. This includes relatively Borel Anosov representations into split simple groups of rank at most three and self-joinings of type-preserving geometrically finite real hyperbolic representations. To prove these results, we develop an intrinsic cusp geometry on the limit set of ρ(Γ) in the Furstenberg boundary and use it to construct a countable Markov coding. We establish exponential moment bounds and uniform distortion estimates for the unbounded vector-valued return cocycle. Using peripheral Cartan growth and Zariski density, we establish the quantitative non-integrability needed for the spectral analysis.
ER -