Analytic Continuation of Resolvent Kernels on Noncompact Symmetric Spaces

Let X = G/K be a symmetric space of noncompact type and let ∆ be the Laplacian associated with a G-invariant metric on X . We show that the resolvent kernel of ∆ admits a holomorphic extension to a Riemann surface depending on the rank of the symmetric space. This Riemann surface is a branched cover of the complex plane with a certain part of the real axis removed. It has a branching point at the bottom of the spectrum of ∆. It is further shown that this branching point is quadratic if the rank of X is odd, and is logarithmic otherwise. In case G has only one conjugacy class of Cartan subalgebras the resolvent kernel extends to a holomorphic function on a branched cover of C with the only branching point being the bottom of the spectrum. Mathematics Subject Classification (2000): 58J50 (11F72)