On Boundedness of Integrable Automorphic Forms in C n

Abstract

We give a necessary and sufficient condition for an integrable automorphic form on a bounded symmetric domain D in Cn to be bounded. The question on the boundedness of integrable automorphic forms on the unit disk in C' has been investigated in [1], [5], [6], [7], [8], [9] and many other papers. Integrable automorphic forms on the bounded homogeneous domains in C' have been considered by Earle [2] and Selberg [10]. In this paper, we shall study the boundedness of integrable automorphic forms on the bounded symmetric domains in Cn. Let D be a bounded symmetric domain in Cn with Bergmen kernel function k(z, w), where z and w represent n-tuples (z,, ... , Zn) and (w,, . .. , wn) respectively. For every holomorphic automorphism g of Aut(D), we have k(z, w) = k(gz, gw)g'(z)g'(w), where g'(z) is the complex Jacobian of the automorphism g. The volume element dm(z) = k(z, z)dz is invariant under the group Aut(D) of all holomorphic automorphisms of D, where dz is the euclidean volume element of D. Let F be a discrete subgroup of Aut(D). We choose a fundamental domain R for F so that aR n D has zero volume. A function f holomorphic on D is said to be an automorphic form of dimension -2q if f(yz)y'(z)q = f(z) for all z in D and y in F. We denote by Aq(F) the space of integrable forms, i.e., the set of all holomorphic automorphic formsf of dimension -2q such that lIflIq = f If(z)I Ik(z, z)1-q/2dm(z) 2 so that all formulas in [2] are valid. c(q) is a certain constant depending only on q. The following theorem is a generalization of a result of Metzger and Rao [7]. Received by the editors January 12, 1979. AMS (MOS) subject classifications (1970). Primary 32N15.