The Meromorphic continuation of the resolvent of the Laplacian on line bundles over ℂH(n)

Abstract

Let G = SU(n,1), K = S(U(n) × U(1)), and for l ∈ Z, let {τ l } l ∈ Z be a one-dimensional K-type and let E l the line bundle over G/K associated to τ l . In this work we prove that the resolvent of the Laplacian, acting on C∞c-sections of E l is given by convolution with a kernel which has a meromorphic continuation to C. We prove that this extension has only simple poles and we identify the images of the corresponding residues with (g, K)-submodules of the principal series representations. We show that for certain values of the parameters these modules are holomorphic (or antiholomorphic) discrete series.