Deformations of Adjoint orbits for semisimple Lie algebras and Lagrangian submanifolds

TL;DR

This paper explores deformations of coadjoint orbits in semisimple Lie algebras, revealing new structures and applications like constructing Lagrangian submanifolds within Hermitian symplectic forms.

Contribution

It introduces a diffeomorphic deformation between classical semisimple coadjoint orbits and those in a semi-direct product, expanding understanding of their geometric structures.

Findings

Established a diffeomorphism between different orbit structures

Demonstrated the existence of Hermitian symplectic forms on these orbits

Constructed Lagrangian submanifolds using the new deformation framework

Abstract

We give a coadjoint orbit's diffeomorphic deformation between the classical semisimple case and the semi-direct product given by a Cartan decomposition. The two structures admit the Hermitian symplectic form defined in a semisimple complex Lie algebra. We provide some applications such as the constructions of Lagrangian submanifolds.