Lagrangian submanifolds of Adjoint semisimple orbits given by real forms

We found some Lagrangian submanifolds of the adjoint semisimple orbit in two cases. For the first, the compact case, also known as the Generalized flag manifolds, we prove that the real flags can be seen as infinitesimally tight Lagrangian submanifolds with respect to the KKS symplectic form and we give a complete classification. And for the second, the complex case, we prove that the orbits of real forms are Lagrangian submanifolds with respect to the Hermitian symplectic form.

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