Classification of Compatible Parabolic Subalgebras for Complex Symmetric Pairs

Abstract

In this paper, we shall recall and arrange the relationship between hyperbolic elements and parabolic subalgebras at first. And then, we shall classify all the compatible parabolic subalgebras containing a $\tau$-stable Borel subalgebra for complex symmetric pairs $(\mathfrak{g},\mathfrak{g}^\tau)$, where $\tau$ is a involutive automorphism of $\mathfrak{g}$.