Almost holomorphic Poincaré series corresponding to products of harmonic Siegel–Maass forms

We investigate Poincaré series, where we average products of terms of Fourier series of real-analytic Siegel modular forms. There are some (trivial) special cases for which the products of terms of Fourier series of elliptic modular forms and harmonic Maass forms are almost holomorphic, in which case the corresponding Poincaré series are almost holomorphic as well. In general, this is not the case. The main point of this paper is the study of Siegel–Poincaré series of degree 2 attached to products of terms of Fourier series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms. We establish conditions on the convergence and nonvanishing of such Siegel–Poincaré series. We surprisingly discover that these Poincaré series are almost holomorphic Siegel modular forms, although the product of terms of Fourier series of harmonic Siegel–Maass forms and holomorphic Siegel modular forms (in contrast to the elliptic case) is not almost holomorphic. Our proof employs tools from representation theory. In particular, we determine some constituents of the tensor product of Harish-Chandra modules with walls.