The universal differential operators on Siegel modular forms

We give an explicit theory of differential operators
acting on Siegel modular forms which preserves automorphy
under the restriction to any diagonal blocks of any size.
For example, we give a generating function including all such operators.
We also show that the Taylor expansion of Siegel modular forms
(or Jacobi forms) w.r.t. off-diagonal-blocks elements
can be recovered by the images of differential operators.
These operators are important in various points, including the calculation
of special values of L functions, or liftings of automorphic forms.