arXiv subjects

Lasse L. Wolf

Publications and source records attributed to Lasse L. Wolf.

6 recordsLinked to original sources

arXiv2026

Spectra of elliptic invariant differential operators on locally symmetric spaces

For a locally symmetric space $Γ\backslash G /K$ we study the spectrum of the invariant differential operators of $G/K$ on $L^2(Γ\backslash G/K)$. This gives a purely analytical proof of the recent theorem of arXiv:2402.02530 that the quasiregular representation $L^2(Γ\backslash G)$ is tempered if the limit cone $\mathcal{L}_Γ$ of $Γ$ is contained in the interior of the Weyl chamber except if $G$ has real rank one or is locally isomorphic to $\mathfrak{sl}_3(\mathbb K)$, $\mathbb K=\mathbb{R},\mathbb{C},\mathbb H$, or $\mathfrak{e}_6^{-26}$.

math.SP↗

arXiv2025

Polyhedral bounds on the joint spectrum and temperedness of locally symmetric spaces

Given a real semisimple connected Lie group $G$ and a discrete subgroup $Γ< G$ we prove a precise connection between growth rates of the group $Γ$, polyhedral bounds on the joint spectrum of the ring of invariant differential operators, and the decay of matrix coefficients. In particular, this allows us to completely characterize temperedness of $L^2(Γ\backslash G)$ in terms of Quint's growth indicator function. As an application of our sharp polyhedral bounds we prove temperedness of $L^2(Γ\backslash G)$ for all Borel Anosov subgroups $Γ$ in higher rank Lie groups $G$ not locally isomorphic to $\mathfrak{sl}_3(\mathbb{K}),\mathbb{K}=\R,\C,\mathbb H,$ or $\mathfrak{e}_{6(-26)}$.

math.RT↗

arXiv2025

On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions

For a large class of symplectic integer matrices, the action on the torus extends to a symplectic $\mathbb{Z}^r$-action with $r\geq 2$. We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the $\mathbb{Z}^r$-action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight $\geq 1/2$ and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight $\geq 1/2$.

math-ph↗

arXiv2024

$L^2$-spectrum, growth indicator function and critical exponent on locally symmetric spaces

In this short note we observe, on locally symmetric spaces of higher rank, a connection between the growth indicator function introduced by Quint and the modified critical exponent of the Poincaré series equipped with the polyhedral distance. As a consequence, we provide a different characterization of the bottom of the $L^2$-spectrum of the Laplace-Beltrami operator in terms of the growth indicator function. Moreover, we explore the relationship between these three objects and the temperedness.

math.SP↗

arXiv2023

Temperedness of locally symmetric spaces: The product case

Let $X=X_1\times X_2$ be a product of two rank one symmetric spaces of non-compact type and $Γ$ a torsion-free discrete subgroup in $G_1\times G_2$. We show that the spectrum of $Γ\backslash X$ is related to the asymptotic growth of $Γ$ in the two direction defined by the two factors. We obtain that $L^2(Γ\backslash G)$ is tempered for large class of $Γ$.

math.SP↗

arXiv2021

Higher rank quantum-classical correspondence

For a compact Riemannian locally symmetric space $Γ\backslash G/K$ of arbitrary rank we determine the location of certain Ruelle-Taylor resonances for the Weyl chamber action. We provide a Weyl-lower bound on an appropriate counting function for the Ruelle-Taylor resonances and establish a spectral gap which is uniform in $Γ$ if $G/K$ is irreducible of higher rank. This is achieved by proving a quantum-classical correspondence, i.e. a 1:1-correspondence between horocyclically invariant Ruelle-Taylor resonant states and joint eigenfunctions of the algebra of invariant differential operators on $G/K$.

math.DS↗