论文标题
$ l^2（γ\ backslash g）$和更高等级的正征本的回火
Temperedness of $L^2(Γ\backslash G)$ and positive eigenfunctions in higher rank
论文作者
论文摘要
令$ g = \ operatoRatorname {so}^\ circ（n，1）\ times \ times \ operatorName {so}^\ circ（n，1）$和$ x = {\ sathbb h}^{n} \ times {\ times {\ times {\ mathb h}^n} $ n \ n \ ge ge 2 $。对于一对$（π_1，π_2）$的非元素凸COCOCOCOCOCOCOCTACT表示，将有限生成的组$σ$ $ \ operatatorName {so}^\ circ（n，1）$，令$γ=（π_1\ timesπ_2）（σ）$。表示$ laplacian的$ l^2 $ -spectrum在$γ\ backslash x $ by $λ_0$上，我们显示： （1）$ l^2（γ\ backslash g）$被纠正，$λ_0= \ frac {1} {2} {2}（n-1）（n-1）^2 $; （2）在$ l^2（γ\ Backslash x）$中没有正laplace eigenfunction。 实际上，（1） - （2）的类似物对任何Anosov子组的$γ$保持至少两个简单的代数组的产品，以及Hitchin子组$γ<\ operatotorname {psl} _d（\ Mathbb r）$，$ d \ d pe 3 $ 3 $。此外，如果$ g $是一个半完美的代数排名群，至少$ 2 $，那么（2）将对任何Anosov子组$γ$的$ G $持有。
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 $(π_1, π_2)$ of non-elementary convex cocompact representations of a finitely generated group $Σ$ into $\operatorname{SO}^\circ(n,1)$, let $Γ=(π_1\times π_2)(Σ)$. Denoting the bottom of the $L^2$-spectrum of the negative Laplacian on $Γ\backslash X$ by $λ_0$, we show: (1) $L^2(Γ\backslash G)$ is tempered and $λ_0=\frac{1}{2}(n-1)^2$; (2) There exists no positive Laplace eigenfunction in $L^2(Γ\backslash X)$. In fact, analogues of (1)-(2) hold for any Anosov subgroup $Γ$ in the product of at least two simple algebraic groups of rank one as well as for Hitchin subgroups $Γ<\operatorname{PSL}_d(\mathbb R)$, $d\ge 3$. Moreover, if $G$ is a semisimple real algebraic group of rank at least $2$, then (2) holds for any Anosov subgroup $Γ$ of $G$.
