POINCARÉ SERIES AND MODULAR FUNCTIONS FOR U(n, 1)

In the theory of automorphic forms, two classes of rank one reductive Lie groups O(n, 1) and U(n, 1) are the important objects. Automorphic forms on O(n, 1) have been intensively studied. In this paper we study the automorphic forms on U(n, 1). We construct infinitely many modular forms and non-holomorphic automorphic forms on U(n, 1) with respect to a discrete subgroup of infinite covolume. More precisely, we obtain the following theorem: Main Theorem. A function f : H C := {Z = (z1, · · · , zn+1) ∈ C : Imzn+1 > ∑n j=1 |zj|} → C is called a nonholomorphic automorphic form attached to the unitary group U(n+ 1, 1) if it satisfies the following three conditions: (1) f is an eigenfunction of the Laplace-Beltrami operator L of U(n+ 1, 1) on