Some Remarks on the Saito Kurokawa Lift for Orthogonal Forms

Duke and Imamoglu showed that the Saito–Kurokawa lift for Siegel modular forms can in an elegant manner be obtained from a converse theorem by Imai using spectral analysis of the hyperbolic Laplacian. In an earlier paper we gave a simplified approach without any appeal to spectral analysis. Here we want to show, that this generalizes to some orthogonal groups of signature (2,m+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(2,m+2)$$\end{document} with m>2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m>2$$\end{document}.


Introduction
There is a well-known correspondence between classical modular forms contained in the Kohnen space and the Maass Spezialschar of Siegel modular forms, called the Saito-Kurokawa-Lift. Kojima [6] and Krieg [3] found an Hermitian analogue of this lifting. Sugano [12] generalized this lifting to orthogonal modular forms. In the case of Siegel modular forms Duke and Imamoglu in [1] gave an alternative proof for the lift using a converse theorem of Imai [2]. In [9] we applied the method of Duke and Imamoglu to the Hermitian case.
Recently we gave simpler proofs for the Saito-Kurokawa lift, see [7,8], where we may skip the analysis of the spectral Koecher-Maass series. The crucial point in our proof is the observation, that the partial Mellin transform of the Saito-Kurokawa-lift of f coincides with a theta lift of f matched with an Eisenstein series, which can easily be evaluated by applying the Rankin Selberg method. The purpose of the present paper is to show how this proof may be generalized to O(2, m + 2) with m > 1.

Siegel Theta Series
Let S be an even integral positive definite matrix of degree m ∈ N with maximal integral lattice Λ 0 = Z m and let Λ 1 = Z × Λ 0 × Z be the even lattice with respect to For later purposes we also introduce the even lattice Λ 2 = Z × Λ 1 × Z with respect to As usual the dual lattice of Λ i is given bŷ The corresponding quadratic forms on V i = R ⊗ Λ i are denoted by Q i = 1 2 S i . We use the common notation Q i [U ] := U t Q i U for any real matrix U . Q 2 has signature (2, m + 2) and Q 1 has signature (1, m + 1). Further remark With this we build a vector-valued theta function Θ(z, P ) = (Θ(r, z, P )) r∈D .

Orthogonal Modular Forms
Siegel modular forms can be regarded as modular forms on O(2, 3), the Hermitian case corresponds to O(2, 4) . Here we shall also consider forms on O(2, m + 2) with m > 2.

The Tube Domain
There is a symmetric space associated to O(Q 2 , R), a model is given by the tube domain In order to understand the action of the orthogonal group on this space one may regard the zero space One finds, that for Z ∈ H and g ∈ O(Q 2 , R) matrix multiplication gives is an automorphy factor, it satisfies the chain rule b(gh, The map g : Z → g Z induces a transitive action of the real orthogonal group O(Q 2 , R) on H∪−H as a group of biholomorphic automorphisms. Define O + (Q 2 , R)} to be the subgroup stabilizing H.

Modular Forms
The orthogonal modular group is now defined as which can be shown to be generated by for all γ ∈ Γ. This space is denoted by M k (Γ). Especially note that Therefore n(ζ) Z = Z + ζ and each holomorphic modular form F (Z) admits an absolutely convergent Fourier development Note, that the Koecher-principle is valid, and therefore we only have summands denotes the level of the lattice Λ 1 . If T ∈ L then for U ∈ Γ 1 also U T ∈ L and because of (2) the Fourier coefficients satisfy the unimodularity property

Saito-Kurokawa Lift
By S(k − m 2 , χ) we denote the space of vector-valued cusp forms of weight k − m 2 that transform according to the representation χ of SL(2, Z) a r (n)e(nz/q).
where  For the proof we shall show that for Y ∈ H m+1 the Mellin transform and then use a converse theorem. The functional equation may result from corresponding equations for the spectral Koecher-Maass series. This was elaborated by Imai in [2] and used by Duke and Imamoglu in [1]. Here we use a shorter argument as we already did in [7,8]. It results from the observation thatF (iY, s) is a theta lift of f matched with a real-analytic Eisenstein series, see (10). The analytic properties of the Eisenstein series, especially its functional equation, carry over toF (Y, s).

Mellin Transform of Unimodular Invariant Fourier Series
Now let us assume, that an absolutely convergent Fourier series on H with the property (4) is given, not necessarily coming from a modular form. Such a Fourier series we call unimodular invariant.
The unimodular invariance implies We further assume, that for Y ∈ H m+1 the partial Mellin transform exists for (s) sufficiently large and Mellin inversion can be applied. We remind the reader that Mellin inversion can be applied if F is continuous and the integral is absolutely convergent in some strip a < (s) < b.

A Converse Theorem
For our proof of the Saito-Kurokawa lift we shall employ the following converse theorem which for the case of Siegel modular forms can be found in [2].
Here we moved the path of integration back to s = c, which is possible becauseF (Y, s) is entire and tends to zero as s → ±∞ uniformly in every vertical strip. The matrix

Rankin-Selberg Integral
The proof of the theorem is complete if we can verify (i)-(iii) of Proposition 5.1. This will be accomplished by the Proposition below which relates the partial Mellin transform to a Rankin-Selberg integral involving the f and the Theta series for some special majorants P Y which owe their relevance to the followig Remark The matrix P Y : where is the real analytic Eisenstein series for SL(2, Z) of weight l ∈ Z. As a function of s the Eisenstein series E l (z, s) has an analytic continuation to the whole complex plane. If l = 0 then ζ(2s)E l (z, s) is an entire function in s and it satisfies the functional equation .
The integral exists for any s which is not a pole of the Eisenstein series, since f is a cusp form and P Y > 0. We now arrive at the announced Proposition. (i) Let T ∈ L + , Y ∈ H m+1 . Then For all s ∈ C with s sufficiently large, we have as an identity of meromorphic functions Proof. Absolute convergence of the defining series is guaranteed from the estimate A(T ) Q a 1 (T ) which in turn follows from well-known estimates for the Fourier coefficients of cusp forms. Also the condition of unimodularity is fulfilled due to the special form of the Fourier coefficients. As in [2], p.910f we conclude that the partial Mellin transform exists for s sufficiently large. The Rankin Selberg integral is evaluated as usual by unfolding the fundamental domain. √ π Γ(s + 1 2 ), the statement of (i) follows for all s with s sufficiently large.
(ii) Since f is a cusp form the Rankin Selberg integral gives a meromorphic continuation forF (Y, s) as function of s to the whole complex plane. It is entire since ζ(2s − k + 1)E −k (z, s − k/2 + 1/2) is an entire function for even positive weight and the poles of Γ(s + 1/2) at s = −1/2 − n for n ∈ N are