Compositions of Automorphic Differential Operators on Siegel Modular Forms

First, we characterize differential operators on Siegel modular forms of degree nm such that the restrictions of the images of the operation to the n n diagonal blocks which consist of the same matrices are again Siegel modular forms of degree n of some different weight.The characterization is given in terms of pluri-harmonic polynomials.Then we show that when n = 1, all such differential operators are obtained by composing two kinds of operators, one which preserves automorphy for the restriction of H m to the diagonals (the product of the upper half planes), and one which preserves automorphy for the restriction from the product of m pieces of upper half planes to the upper half plane embedded diagonally.We sometimes identify Sp(n, R) with the image of this embedding.Then the action of Sp(n, R) on H n and the action on H n 1 m H nm are equivariant.