New analytic problems over imaginary quadratic number fields

In the present article we shall ponder the possibility of extending the investigation developed in the book [11] to imaginary quadratic number fields. We shall be led in a natural way to the spectral theory of the hyperbolic upper half-space, which if looked at from our standpoint will turn out to be not a sheer counterpart of the 2-dimensional case but a fascinating subject raising new arithmetic and analytic problems, the roots of which can be attributed to Hecke [2] and Maass [6]. 1991 Mathematics Subject Classification: 11M06.