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We study theta lifts for   . 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. For the regularized theta series we compute the theta lift for cusp forms. The regularized lift also gives a correspondence for non-cusp forms such as Eisenstein series. Also we obtain the spectral expansion of the theta series in either of its variables. As an application we prove a three dimensional analogue of Katok–Sarnak's correspondence using the Selberg transform.

Regularized theta lift and formulas of Katok–Sarnak type