Regularized theta lifts and Niebur-type Poincaré series on n -dimensional hyperbolic space

Abstract

We study theta liftings from H n to H 2 and vice versa. The theta-lift is realized via an integral transform with a Siegel theta series as kernel function. Since this Siegel theta series fails to be square integrable it has to be regularized. The regularization is obtained by applying a suitable differential operator built from the Laplacian. The Siegel theta series is seen to be related to an automorphic Selberg kernel function on H n and therefore the Selberg transform applies to compute the Fourier coefficients of the lift. This gives formulas of Katok–Sarnak type for Fourier coefficients of positive as well as for negative index, involving geodesic cycles. As an application we can lift Poincaré series of Niebur type. The lifted series are again Poincaré series built with Whittaker functions.