Invariant differential operators on Siegel-Jacobi space and Maass-Jacobi forms

In this talk I will discuss differential operators on the Siegel-Jacobi space invariant under the
natural action of the Jacobi group, and using these invariant differential operators we study
Maass-Jacobi forms.
The Siegel-Jacobi space is a very important non-reductive homogeneous space
in the aspects of arithmetic and geometry. I review the works of Hans Maass and
Goro Shimura about invariant differential operators on the Siegel space roughly.
I will present my results on invariant  differential operators on the
Siegel-Jacobi space. Also I present some examples of explicit  invariant
differential operators on the Siegel-Jacobi space. In the  end of my talk, I
will propose several important problems that must be investigated in  the
future.