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

We undertake a detailed study of the lowest weight modules for the Hermitian symmetric pair (G,K), whereG=Sp4(R) andK is itsmaximal compact subgroup. In particular, we determineK-types and composition series, andwrite down explicit differential operators that navigate all the highest weight vectors of such a module starting from the unique lowestweight 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 Siegelmodular forms of degree 2. Further, by explicating the algebraic structure of the relevant space of n-finite automorphic forms, we areable toprove astructure theorem for the space of nearly holomorphic vector-valued Siegel modular forms of (arbitrary) weight detl symm with respect to an arbitrary congruence subgroup of Sp4(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 detl' symm' with (l',m') varying over a certain set. The structure theorem for the space of all modular forms is similar, except thatwemay nowhave an additional component coming from certain nearly holomorphic forms of weight det³ symm' 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. [ABSTRACT FROM AUTHOR]