The meromorphic continuation of the resolvent of the Laplacian on line bundles over \(\mathbb C H(n)\).
Let \(G=SU(n,1)\), \(K=S(U(n)\times U(1))\), and let \(\{\tau_l\}_{l\in\mathbb{Z}}\) be the one-dimensional \(K\)-types and \(E_l\) be the line bundle over \(G/K\) associated to \(\tau_l\). Generalizing (part of) the results of a previous paper by the same author and \textit{R. Miatello} [Proc. Am. Math. Soc. 128, 1221--1229 (2000; Zbl 0941.22005)] for the case of \(l=0\) (i.e.~when \(E_l\) is the trivial bundle), the author proves that the resolvent of the Laplacian acting on the \(C^\infty\) sections of \(E_l\) is given by a convolution with a kernel (given quite explicitly in terms of certain hypergeometric functions) which has a meromorphic continuation to the whole complex plane. This continuation has only simple poles at the points \(\nu^\pm_{k,l}=\pm| l| -n-2k\), where \(k\in\mathbb{Z}\) and \(k\geq0\) for \(\nu^-_{k,l}\) and \(0\leq 2k\leq| l| -n\) for \(\nu^+_{k,l}\). Further, the images of the residues are \((\mathfrak g,K)\)-submodules which are finite-dimensional in the case of \(\nu^-_{k,l}\), while for \(\nu^+_{k,l}\) they are equivalent to the holomorphic discrete series representations. A~more detailed description of the simplest case \(n=1\) is given separately in the final section.
scientific article; zbMATH DE number 2101067