Temperedness of $L^2(\Gamma\backslash G)$ and positive eigenfunctions in higher rank

TL;DR

This paper proves that for certain higher rank groups, the space of square-integrable functions is tempered with a specific spectral bottom, and there are no positive eigenfunctions in the associated quotient spaces, extending these results to various algebraic group subgroups.

Contribution

The paper establishes new results on the temperedness of $L^2(\Gammaackslash G)$ and the non-existence of positive eigenfunctions for higher rank groups and their subgroups, generalizing previous spectral theory.

Findings

$L^2(\Gammaackslash G)$ is tempered with $\lambda_0=rac{1}{2}(n-1)^2$

No positive Laplace eigenfunctions exist in $L^2(\Gammaackslash X)$

Results extend to Anosov and Hitchin subgroups in higher rank algebraic groups

Abstract

Let $G=\operatorname{SO}^\circ(n,1) \times \operatorname{SO}^\circ(n,1)$ and $X={\mathbb H}^{n}\times {\mathbb H}^{n}$ for $n\ge 2$. For a pair $(\pi_1, \pi_2)$ of non-elementary convex cocompact representations of a finitely generated group $\Sigma$ into $\operatorname{SO}^\circ(n,1)$, let $\Gamma=(\pi_1\times \pi_2)(\Sigma)$. Denoting the bottom of the $L^2$-spectrum of the negative Laplacian on $\Gamma\backslash X$ by $\lambda_0$, we show: (1) $L^2(\Gamma\backslash G)$ is tempered and $\lambda_0=\frac{1}{2}(n-1)^2$; (2) There exists no positive Laplace eigenfunction in $L^2(\Gamma \backslash X)$. In fact, analogues of (1)-(2) hold for any Anosov subgroup $\Gamma$ in the product of at least two simple algebraic groups of rank one as well as for Hitchin subgroups $\Gamma<\operatorname{PSL}_d(\mathbb R)$, $d\ge 3$. Moreover, if $G$ is a semisimple real algebraic group of rank at least $2$, then (2) holds for any Anosov subgroup $\Gamma$ of $G$.