Holomorphic projection for $\mathop{Sp}_2(\mathbb R)$ -- the case of weight $(4,4)$

TL;DR

This paper develops a method for holomorphic projection of automorphic forms on the symplectic group Sp_2(R) at weight (4,4), using non-holomorphic Poincaré series, Casimir operators, and spectral analysis.

Contribution

It introduces a new approach to holomorphic projection for Sp_2(R) at weight (4,4) via analytic continuation of Poincaré series and spectral theory.

Findings

Constructed non-holomorphic Poincaré series of exponential type for Sp_2(R).

Achieved analytic continuation of these series at weight (4,4).

Expressed holomorphic projection through Fourier coefficients and Sturm's operator.

Abstract

We define non-holomorphic Poincar\'e series of exponential type for symplectic groups $\mathop{Sp}_m(\mathbb R)$ and continue them analytically in case $m=2$ for the small weight $(4,4)$. For this we construct certain Casimir operators and study the spectral properties of their resolvents on $L^2(\Gamma\backslash \mathop{Sp}_2(\mathbb R))$. Using the holomorphically continued Poincar\'e series, the holomorphic projection is described in terms of Fourier coefficients using Sturm's operator.