A Characterization of Planar Mixed Automorphic Forms

We characterize the functional space of the planar mixed automorphic forms with respect to an equivariant pair and given lattice as the image of the Landau automorphic forms (involving special multiplier) by an appropriate isomorphic transform. 1. Introduction Mixed automorphic forms of type arise naturally as holomorphic forms on elliptic varieties [1] and appear essentially in the context of number theory and algebraic geometry. Roughly speaking, they are a class of functions defined on a given (Hermitian symmetric) space and satisfying a functional equation of type for every and . Here, ; is an automorphic factor associated to an appropriate action of a group on , and is an equivariant pair for the data . Such notion was introduced by Stiller [2] and extensively studied by Lee in the case of being the upper half-plane. They include the classical ones as a special case. Nontrivial examples of them have been constructed in [3, 4]. We refer to [5] for an exhaustive list of references. In this paper, we are interested in the space of mixed automorphic forms defined on the complex plane with respect to a given lattice in and an equivariant pair . We find that is isomorphic to the space of Landau automorphic forms [6], of “weight” with respect to a special pseudocharacter defined on and given explicitly through (5.3) below. The crucial point in the proof is to observe that the quantity is in fact a real constant independent of the complex variable . The exact statement of our main result (Theorem 5.1) is given and proved in Section 5. In Sections 2 and 3, we establish some useful facts that we need to introduce the space of planar mixed automorphic forms . We have to give necessary and sufficient condition to ensure the nontriviality of such functional space. In Section 4, we introduce properly the function that serves to define the pseudocharacter . 2. Group Action Let be the semidirect product group of the unitary group and the additive group . acts on the complex plane by the holomorphic mappings ; , , so that can be realized as Hermitian symmetric space . By a -equivariant pair , we mean that is a -endomorphism and is a compatible mapping, that is, Now, for given real numbers , and an equivariant pair , we define to be the complex-valued mapping where ; is the “automorphic factor” given by Here and elsewhere, denotes the imaginary part of the complex number and the usual Hermitian scalar product on . Thus, one can check the following. Proposition 2.1. The mapping satisfies the chain rule where is the real-valued function defined on by Proof. For every