On the variety of Lagrangian subalgebras, I

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 variet y L of Lagrangian subalgebras carries a natural Poisson structure II. We de termine the irreducible components of L, and we show that each irreducible component is a smooth fiber bundle over a generalized flag variety, and tha t the fiber is the product of the set of real points of a De Concini-Proces i compactification and a connected component of a real orthogonal group. We study some properties of the Poisson structure II and show that L contains many interesting Poisson submanifolds. (C) 2001 Editions scientifiques et medicales Elsevier SAS.