On the variety of Lagrangian subalgebras

TL;DR

This paper investigates the geometric and Poisson structures of Lagrangian subalgebras in semisimple Lie algebras, revealing their fiber bundle structure over flag varieties and properties of their Poisson structure.

Contribution

It characterizes the irreducible components of the variety of Lagrangian subalgebras and describes their fiber bundle structure and Poisson properties.

Findings

Irreducible components are smooth fiber bundles over flag varieties.

Each fiber is a product of real points of a De Concini-Procesi compactification and a homogeneous space.

The Poisson structure contains many interesting submanifolds.

Abstract

We study Lagrangian subalgebras of a semisimple Lie algebra with respect to the imaginary part of the Killing form. We show that the variety $\Lagr$ of Lagrangian subalgebras carries a natural Poisson structure $\Pi$. We determine the irreducible components of $\Lagr$, and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and that the fiber is the product of the real points of a De Concini-Procesi compactification and a compact homogeneous space. We study some properties of the Poisson structure $\Pi$ and show that it contains many interesting Poisson submanifolds.