Hamiltonian actions and Lagrangian homogeneous submanifolds

Abstract

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired to the recent paper \cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is given and we also describe under which condition a Lagrangian orbit is isolated. If $M$ is a compact K\"ahler manifold we give a necessary and sufficient condition to an isometric action admits a Lagrangian orbit. Then we investigate Lagrangian homogeneous submanifolds on the symplectic cut and on the symplectic reduction. As an application of our results, we give new examples of Lagrangian homogeneous submanifolds on the blow-up at one point of the complex projective space and on the weighted projective spaces. Finally, applying Proposition \ref{slice} that we may call \emph{Lagrangian slice theorem} for group acting with a fixed point, we give new examples of Lagrangian homogeneous submanifolds on irreducible Hermitian symmetric spaces of compact and noncompact type.