论文标题
缺少高级局部对称空间的主要特征值
Absence of principal eigenvalues for higher rank locally symmetric spaces
论文作者
论文摘要
鉴于无限体积的几何有限双曲线表面是帕特森的经典结果，即正laplace-beltrami操作员没有$ l^2 $ -eigenValues $ \ geq 1/4 $。在本文中，我们证明了该结果的概括，即在Riemannian本地对称空间上通勤差分运算符的代数的联合$ l^2 $ eigenvalues $γ\ backslash g/k $更高的等级。我们在大地测量和SATAKE压缩的$γ$ ACTION上得出动态假设，这意味着没有相应的主特征值。满足这些假设的大量示例是Anosov子组的非紧凑型商。
Given a geometrically finite hyperbolic surface of infinite volume it is a classical result of Patterson that the positive Laplace-Beltrami operator has no $L^2$-eigenvalues $\geq 1/4$. In this article we prove a generalization of this result for the joint $L^2$-eigenvalues of the algebra of commuting differential operators on Riemannian locally symmetric spaces $Γ\backslash G/K$ of higher rank. We derive dynamical assumptions on the $Γ$-action on the geodesic and the Satake compactifications which imply the absence of the corresponding principal eigenvalues. A large class of examples fulfilling these assumptions are the non-compact quotients by Anosov subgroups.
