Lagrangian submanifolds of Adjoint semisimple orbits given by real forms

TL;DR

This paper investigates Lagrangian submanifolds within adjoint semisimple orbits, providing classifications and demonstrating their properties in both compact and complex cases using symplectic forms.

Contribution

It offers a complete classification of real flags as Lagrangian submanifolds in compact cases and proves that real form orbits are Lagrangian in the complex case.

Findings

Real flags are infinitesimally tight Lagrangian submanifolds in compact cases.

Orbits of real forms are Lagrangian with respect to the Hermitian symplectic form.

Complete classification of these Lagrangian submanifolds.

Abstract

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.