Almost holomorphic Poincare series corresponding to products of harmonic Siegel-Maass forms

TL;DR

This paper studies Siegel-Poincaré series formed from products of Fourier terms of harmonic Siegel-Maass forms and holomorphic Siegel modular forms, revealing they are almost holomorphic despite the non-almost holomorphic nature of the products.

Contribution

It establishes convergence and nonvanishing conditions for these Poincaré series and shows they are almost holomorphic Siegel modular forms, a surprising result in contrast to the elliptic case.

Findings

Poincaré series are almost holomorphic Siegel modular forms.

Conditions for convergence and nonvanishing are established.

Representation theory tools are employed in the proof.

Abstract

We investigate Poincar\'e 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 Poincar\'e series are almost holomorphic as well. In general this is not the case. The main point of this paper is the study of Siegel-Poincar\'e 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-Poincar\'e series. We surprisingly discover that these Poincar\'e 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.