Angela Pasquale

Resonances for the Laplacian on Riemannian symmetric spaces

Let D be the Laplacian on a Riemannian symmetric space of the noncompact type X=G/K, and let s(D) denote its spectrum. The resolvent R(z)=(D-z)^{-1} is a holomorphic function on C \ s(D), with values in the space of bounded operators on L^2(X). We study the meromorphic continuation of R as a distribution valued map on a Riemann surface above C \ s(D). If such a meromorphic continuation is possible, then the poles of the meromorphically extended resolvent are called the resonances. When all Cartan subgroups of G are conjugate, then there are no resonances. In other examples the resonances exist and can be explicitly determined. They are linked to the spherical principal series representations of G. This talk is based on joint works with Joachim Hilgert (Universität Paderborn) and Tomasz Przebinda (University of Oklahoma).