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

Cynthia E. Will

Vol. 209 (2003), No. 1, 157–173

DOI: 10.2140/pjm.2003.209.157

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.

Milestones

Received: 8 June 2000

Revised: 10 May 2001

Published: 1 March 2003