Abstract

Let G = SU(n,1), K = S(U(n) × U(1)), and for l in Z, let {τ_l}_{l in 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.

Authors