On the variety of Lagrangian subalgebras, I

Abstract

We study subalgebras of a semi-simple Lie algebra which are Lagrangian with respect to the imaginary part of the Killing form. We show that the variety L of Lagrangian subalgebras carries a natural Poisson structure Π. We determine the irreducible components of L , 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 set of real points of a De Concini–Procesi compactification and a connected component of a real orthogonal group. We study some properties of the Poisson structure Π and show that L contains many interesting Poisson submanifolds.