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

Abstract

We define non-holomorphic Poincaré series of exponential type for symplectic groups ${\mathrm{Sp}}_m(\mathbb{R})$ 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 $L^2(\Gamma \backslash {\mathrm{Sp}}_2(\mathbb{R}))$. Using the holomorphically continued Poincaré series, the holomorphic projection is described in terms of Fourier coefficients using Sturm's operator.