On Poincaré series of exponential type on Sp\(_ 2\)
Let \(\Gamma_ 2=\text{Sp}_ 2(\mathbb{Z})\) denote the Siegel modular group of degree 2. If \(T\) is a positive definite half-integral \(2\times 2\) matrix, the Poincaré series of exponential type \[ P_{k,T}(Z)= \sum_{M={AB\choose CD}:{I^*\choose 0I}\backslash\Gamma_ 2} \text{det}(CZ+D)^{-k} e^{2\pi i\text{ tr}(TM\langle Z\rangle)}, \] \(k>4\) even, is known to be a Siegel cusp form of weight \(k\). If \(k\geq 8\) is even, the author can describe a representation of \(P_{k,T}\) as an infinite linear combination of Poincaré series on the Jacobi group \(\Gamma^ J_ 1= \text{SL}_ 2(\mathbb{Z})\ltimes \mathbb{Z}^ 2\). This yields the Fourier-Jacobi expansion of \(P_{k,T}\). As an application an elementary formula for the Fourier coefficients of \(P_{k,T}\) is derived. Moreover the adjoint with respect to the Petersson inner product of the maps sending a Siegel cusp form onto its various Fourier-Jacobi coefficients can be described.
scientific article; zbMATH DE number 492599