On Koecher-Maass series of Siegel modular forms
Here we see a new proof that the Koecher-Maass series of a Siegel modular form \(f\) (for the full integral symplectic group) gives rise to a function \(D^*_f(s)\) having an analytic continuation to the whole complex plane and obeying a functional equation. This is a classical result, but the proof offered here is by constructing a holomorphic kernel function for \(f\mapsto D^*_f(s)\). More precisely the authors exhibit a function \(F_k(Z;s)\) (for \(s\in{\mathbb C}\) and \(Z\) in the Siegel upper half-plane) such that in a suitable region \[ \left\langle F_k(\cdot;s), f^*\right\rangle=C_{n,k}D^*_f(s), \] where \(f\) is a weight \(k\) form for \(\text{ Sp}_n\), \(f^*\) is obtained by taking the complex conjugate of the Fourier coefficients and \(\langle\cdot,\cdot\rangle\) is the Petersson scalar product. The results are deduced from analogous properties of \(F_k\), which are fairly easy consequences of its definition by a series. The proof seems on the face of it to be very different from the classical proof of these results. There is a mild restriction: at least as given here, the new proof is only valid for \(k>3(n+1)\) (and even).
scientific article; zbMATH DE number 2075722