Absence of Principal Eigenvalues for Higher Rank Locally Symmetric Spaces

Abstract

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 ge 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 Gamma backslash G/K of higher rank. We derive dynamical assumptions on the Gamma -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.