Lowest weight modules of Sp4(R) and nearly holomorphic Siegel modular forms

Abstract

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair $(G,K)$, where $G={\mathrm{Sp}}_4(\mathbb{R})$ and K is its maximal compact subgroup. In particular, we determine K-types and composition series, and write down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowest weight vector. By rewriting these operators in classical language, we show that the automorphic forms on G that correspond to the highest weight vectors are exactly those that arise from nearly holomorphic vector-valued Siegel modular forms of degree 2.

Further, by explicating the algebraic structure of the relevant space of $\mathfrak{n}$-finite automorphic forms, we are able to prove a structure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight ${det}^{\ell }{\mathrm{sym}}^m$ with respect to an arbitrary congruence subgroup of ${\mathrm{Sp}}_4(\mathbb{Q})$. We show that the cuspidal part of this space is the direct sum of subspaces obtained by applying explicit differential operators to holomorphic vector-valued cusp forms of weight ${det}^{{\ell }^{\prime }}{\mathrm{sym}}^{m^{\prime }}$ with $({\ell }^{\prime },m^{\prime })$ varying over a certain set. The structure theorem for the space of all modular forms is similar, except that we may now have an additional component coming from certain nearly holomorphic forms of weight ${det}^3{\mathrm{sym}}^{m^{\prime }}$ that cannot be obtained from holomorphic forms.

As an application of our structure theorem, we prove several arithmetic results concerning nearly holomorphic modular forms that improve previously known results in that direction.