## Tohoku Mathematical Journal

### Hamiltonian actions and homogeneous Lagrangian submanifolds

Leonardo Biliotti

#### Abstract

We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired by recent results, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is given. Also, we describe under which condition a Lagrangian orbit is isolated. If $M$ is a compact Kähler manifold, we give a necessary and sufficient condition for an isometric action to admit a Lagrangian orbit. Then we investigate homogeneous Lagrangian submanifolds on the symplectic cut and on the symplectic reduction. As an application of our results, we exhibit new examples of homogeneous Lagrangian submanifolds on the blow-up at one point of the complex projective space and on the weighted projective spaces. Finally, applying our result which may be regarded as Lagrangian slice theorem for a Hamiltonian group action with a fixed point, we give new examples of homogeneous Lagrangian submanifolds on irreducible Hermitian symmetric spaces of compact or noncompact type.

#### Article information

Source
Tohoku Math. J. (2), Volume 59, Number 4 (2007), 603-616.

Dates
First available in Project Euclid: 6 January 2008

https://projecteuclid.org/euclid.tmj/1199649876

Digital Object Identifier
doi:10.2748/tmj/1199649876

Mathematical Reviews number (MathSciNet)
MR2404207

Zentralblatt MATH identifier
1137.53020

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index
Secondary: 53D20: Momentum maps; symplectic reduction

#### Citation

Biliotti, Leonardo. Hamiltonian actions and homogeneous Lagrangian submanifolds. Tohoku Math. J. (2) 59 (2007), no. 4, 603--616. doi:10.2748/tmj/1199649876. https://projecteuclid.org/euclid.tmj/1199649876

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