IKEDA TYPE CONSTRUCTION OF CUSP FORMS (Modular forms and automorphic representations)

This is a survey of results on the construction of holomorphic cusp forms on tube domains originally initiated by Ikeda [9].Besides a survey it includes conjectures and possible applications of our work [19]. INTRODUCTIONThere are five simple tube domains (cf. [6]).They are of the form $\mathfrak{D}=\{Z=X+iY|X\in$ $\mathbb{R}^{n},$ $Y\in C\}$ , where $C$ is a self-adjoint homogeneous cone in $\mathbb{R}^{n}$ .Let $G$ be (the real points of) the simply connected, simple real algebraic group which acts transitively on $\mathfrak{D}$ .We list the group $G$ and the cone $C$ :(1) $Sp_{2n}$ (rank $n$ ) $;n\cross n$ positive definite matrices over $\mathbb{R}$ ;(2) $SU(n, n);n\cross n$ positive definite hermitian matrices over $\mathbb{C}$ ;(3) $SU(2n, H)=Spin^{*}(4n);n\cross n$ positive definite hermitian matrices over $H$ (quater- nions);(4) $SO(2, n)^{0}$ ; the cone in $\mathbb{R}^{n+1}$ of $(x_{0}, \ldots, x_{n})$ with $x_{0}>(x_{1}^{2}+\cdots+x_{n}^{2})^{\frac{1}{2}}$ ;(5) $E_{7,3};3\cross 3$ positive definite hermitian matrices over $\mathfrak{C}$ (Cayley numbers).It is an important problem to explicitly construct holomorphic cusp forms on $\mathfrak{D}$ with respect to $G(\mathbb{Z})$ (we will call such a modular form on $\mathfrak{D}$ "a level one form In particular, we focus on the lifting from normalized Hecke cusp eigenforms on the complex upper half-plane $\mathbb{H}$ with respect to $SL_{2}(\mathbb{Z})$ to holomorphic cusp forms on $\mathfrak{D}.$Ikeda [9] (see also [8]) gave $a$ (functorial) construction of Siegel cusp forms of weight $n+k,$ $n\equiv k$ mod2 (so that $n+k$ is even) for $Sp_{4n}$ from normalized Hecke eigenforms in $S_{2k}(SL_{2}(\mathbb{Z}))$ which has been conjectured by Duke and Imamoglu (Independently Ibukiyama formulated a conjecture in terms of Koecher-Maass series).He made use of the uniform property of the Fourier coefficients of Siegel Eisenstein series for $Sp_{4n}$ and together with various deep facts established in [9] to prove Duke-Imamoglu conjecture.When $n=1$ , it is nothing but a Saito-Kurokawa lift.Since then, his construction was generalized to unitary

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