The variety of Lagrangian subalgebras of real semisimple Lie algebras

The purpose of our work is to generalize a result of Evens and Lu cite{rfEvensLu2} from the complex case to the real one. In cite{rfEvensLu2}, Evens and Lu considered the variety $L$ of complex Lagrangian subalgebras of $g mg,$ where $g$ is a complex semisimple Lie algebra. They described irreducible components, and showed that all irreducible components are smooth. In particular, some irreducible components are De Concini-Procesi compactifications of the adjoint group $G$ associated to $g,$ and its geometry at infinity plays a role in understanding $L$ and in applications in Lie theory cite{rfEvensLuPoisson}, cite{rfEvensLuThom}. In this work we consider an analogous problem, where $ g$ is a real semisimple Lie algebra. We describe the irreducible components of $ L,$ the real algebraic variety of Lagrangian subalgebras of $ g m g$. Let $g$ be the complexification of $ g.$ Denote by $sigma$ the complex conjugation on $g$ with respect to $ g.$ Let $G$ be the adjoint group of $g.$ Then $sigma$ can be lifted to an antiholomorphic involutive automorphism $sigma: G ightarrow G,$ which we denote by $sigma$ too. The set of $sigma$-fixed points $G^{sigma}={gin G : sigma(g) =g}$ is a real algebraic group with Lie algebra $ g.$ We study $G^{sigma}$ using a result of Steinberg cite{Steinberg} and the Atlas project software cite{atlas}. Our results also give a description of a compactification of $G^{sigma}.$ Although our methods are similar to the methods of cite{rfEvensLu2}, the real structure introduces some additional difficulties.