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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.

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