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

Abstract

Let $G={\mathrm{SO}}^{\circ }(n,1)\times {\mathrm{SO}}^{\circ }(n,1)$ 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 ${\mathrm{SO}}^{\circ }(n,1)$, 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 $\Gamma <{\mathrm{PSL}}_d(\mathbb{R})$, $d\ge 3$. Moreover, if $G$ is a semisimple real algebraic group of rank at least $2$, then (2) holds for any Anosov subgroup $\Gamma$ of $G$.

38 pages, Final version, To appear in Communications of the AMS