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

Let $G=\operatorname {SO}^\circ (n,1) \times \operatorname (n,1)$ 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 finitely generated group $\Sigma$ into $\operatorname (n,1)$, let $\Gamma =(\pi _1\times _2)(\Sigma )$. Denoting the bottom $L^2$-spectrum negative Laplacian on {\setminus } X$ by $\lambda _0$, we show: $L^2(\Gamma G)$ is tempered _0=\frac {1}{2}(n-1)^2$; There exists no positive Laplace eigenfunction in X)$. In fact, analogues (1)-(2) hold any Anosov subgroup $\Gamma$ product at least two simple algebraic groups rank one as well Hitchin subgroups <\operatorname {PSL}_d(\mathbb {R})$, $d\ge 3$. Moreover, if $G$ semisimple real $2$, then (2) holds $G$.