On Poincaré series on \(Sp_ m\) (Q1922569)

The purpose of this article is to compute the Fourier-Jacobi expansion of (holomorphic) cuspidal Poincaré series of exponential type on the Siegel modular group \(\Gamma_m= \text{Sp} (m, \mathbb{Z})\). This calculation generalizes \textit{W. Kohnen}'s result for \(m=2\) [Abh. Math. Semin. Univ. Hamb. 63, 283-297 (1993; Zbl 0790.11038)]. More precisely, given an even integer \(k> 2m\) and \(T\) a positive definite symmetric half-integral \((m, m)\)-matrix, denotes by \(P_{k, T}\) the \(T\)th Poincaré series of weight \(k\) on \(\Gamma_m\). Also fix a decomposition \(m= n+ r\) of \(m\) into a sum of two integers \(n,r> 0\). The main result (Theorem 3) of this paper is that the Fourier-Jacobi coefficients of \(P_{k, T}\) can be written as an infinite linear combination of Jacobi-Poincaré series on the Jacobi groups \(\Gamma^J_{n,r}\) and \(\Gamma^J_{r,n}\). An application to an explicit description (in case \(n=r\)) of the adjoints with respect to the Petersson scalar products of the maps which send a Siegel modular form to its various Fourier-Jacobi coefficients is given.