A characterization of planar mixed automorphic forms

We characterize the space of the so-called planar mixed automorphic forms of type $(\nu,\mu)$ with respect to an equivariant pair $(\rho,\tau)$ as the image, by an appropriate transform, of the usual (Landau) automorphic forms involving special multiplier.


INTRODUCTION
The notion of mixed automorphic forms was introduced by Stiller [7] and extensively studied by M.H. Lee (see [5] and the references therein). They appears essentially in the context of number theory and algebraic geometry and arise naturally as holomorphic forms on elliptic varieties [4]. Mixed automorphic forms include classical ones as a special case and non trivial examples of them has being constructed in [1,6]. In this paper, we deal with the space planar mixed automorphic forms and we show that it can be connected to the space of Landau automorphic forms [2], by an explicit and special transform (Theorem 3.1).
Let C be the complex plane endowed with its usual hermitian scalar product z, w = zw, and T be the unitary group, T = {λ ∈ C; |λ| = 1}. Consider the semidirect product group G = T ⋊ C operating on C by the holomorphic mappings g · z = az + b for g = (a, b) ∈ G. By equivariant pair (ρ, τ), we mean that ρ is a G-endomorphism and τ : C → C a compatible mapping such that Associated to such (ρ, τ) and given uniform lattice Γ in C, we consider the vector space M ν,µ τ (C) of Γ-mixed automorphic forms of type (ν, µ). They are smooth complex-valued functions F on C satisfying the functional equation where ν, µ are non negative real numbers and j α ; α ∈ R, is defined by Here and elsewhere ℑz denotes the imaginary part of the complex number z. We assert that the space M ν,µ τ (C) is isomorphic to the space of automorphic forms F ∈ C ∞ (C), i.e., such that The pseudo-character χ τ is defined on Γ by where the involved function ϕ ν,µ τ satisfies a first order differential equation as in Proposition 2.3 below.
Our main result (Theorem 3.1) is stated and proved in Section 3. To do this, we have to ensure first the nontrivially of the space M ν,µ τ (C) and to introduce properly the function ϕ ν,µ τ (Section 2). The crucial point in the proof of Theorem 3.1 is to observe that the weight B ν,µ τ given by B is indeed a real constant independent of the complex variable z. As immediate application of the obtained characterization, one can deduce easily some concrete spectral properties of an appropriate invariant Laplacian acting on M ν,µ τ (C) (see [3] for more details).

ON THE SPACE
For given real numbers ν, µ > 0 and given equivariant pair and we perform the vector space of mixed automorphic forms of type (ν, µ), 2) Then, one can check the following Then, the mapping J ν,µ ρ,τ satisfies the chain rule

5)
for varying g ∈ G, define then a projective representation of the group G on the space of C ∞ functions on C. While the assertion ii) shows that M ν,µ τ (C) can be realized as the space of cross sections on a line bundle over the complex torus C/Γ.
Proof. For every g, g ′ ∈ G and z ∈ C, we have Next, one can sees that the automorphic factor j α (·, ·) satisfies This gives rise to The proof of ii) can be handled in a similar way as in [2] making use of (2.4) combined with the equivariant condition (1.1).
In order to prove the main result of this paper, we need to introduce the function ϕ ν,µ τ .