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

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 $G$.