Orbits on Lagrangian Grassmanian

The purpose of this note is to give a classification of the orbital structure of certain reductive group actions on the Lagrangian Grassmanian. The groups under consideration are $Sp \times Sp$ and $GL$. The classification of $Sp \times Sp$ orbits is well-known and the classification of $GL$-orbits is not known. We also compute the isotropy group. Some of the results here will be used to analyze the representations of the universal covering of the symplectic group on the Shilov boundary.