Abstract
For a complex semisimple Lie algebra g with Hermitian real form gR=kR+pR, there exists a positive system of roots such that the adjoint k-representation on p stabilizes the positive and negative root spaces. In this article, we extend this result to contragredient Lie superalgebras g, and study the number of irreducible components of the k-representation. We also discuss the complex structure on gR/kR.