Hermitian real forms of contragredient Lie superalgebras

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