# Infinitesimally Tight Lagrangian Orbits

## TL;DR

This paper studies special Lagrangian orbits in symplectic manifolds, introducing the concept of infinitesimally tight orbits and exploring their intersection properties through examples in complex flag manifolds and cotangent bundles.

## Contribution

It introduces the notion of infinitesimally tight Lagrangian orbits and analyzes their intersection theory, providing new examples in various symplectic manifolds.

## Key findings

- Examples of Lagrangian orbits in complex flag manifolds and cotangent bundles.
- Introduction of the concept of infinitesimally tight orbits.
- Development of intersection theory for these orbits.

## Abstract

We describe isotropic orbits for the restricted action of a subgroup of a Lie group acting on a symplectic manifold by Hamiltonian symplectomorphisms and admitting an Ad*-equivariant moment map. We obtain examples of Lagrangian orbits of complex flag manifolds, of cotangent bundles of orthogonal Lie groups, and of products of flags. We introduce the notion of infinitesimally tight and study the intersection theory of such Lagrangian orbits, giving many examples.