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

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