Maass Spaces and a Characterization of Images of Ikeda Liftings

For an arbitrary even genus n we show that the subspace of Siegel cusp forms of weight k + n/2 generated by Ikeda lifts of elliptic cusp forms of weight 2k can be characterized by certain linear relations among Fourier coefficients. This generatizes the work of Kohnen and Kojima. We investigate the analogous subspaces of hermitian and quaternionic cusp forms. Introduction Ikeda [7] constructed a lifting from S2k(SL2(Z)) to Sk+n/2(Spn(Z)) for an integer k and an even integer n such that k ≡ n/2 (mod 2). He constructed a lifting that associates to an elliptic cusp form a hermitian cusp form (cf. [8]). The author constructed a lifting that associates to an elliptic cusp form a quaternionic cusp form (cf. [23]). The purpose of this paper is to give a characterization of the images of these liftings by certain linear relations among Fourier coefficients. Let us describe our results. The letter F stands for Q, an imaginary quadratic field K or a definite quaternion algebra H over Q. Let ι be the attached involution, i.e., the identity map, the complex conjugate or the main involution accordingly as F = Q, K or H . Put x = (xji) for x = (xij) ∈ Mn(F ). Fix a maximal order R of F . Let Gn be an algebraic group defined over Q, the group of Q-rational points of which is given by Gn(Q) = { α ∈ SL2n(F ) ∣∣ α ( 0 −1n 1n 0 ) α = ( 0 −1n 1n 0 )} . Put Sn(O) = {x ∈ Mn(O) | x = x} for an involutive algebra O. The archimedean part Gn(R) acts transitively on the upper half-space Hn = {Z = X + √ −1Y ∈ Sn(F)⊗R C | X ∈ Sn(F), 0 < Y ∈ Sn(F)},