Double Vogan diagrams and irreducible pseudo-Hermitian symmetric spaces

Abstract

Abstract. Let G be a real simple Lie group with Lie algebra g. We consider additional structures on the Dynkin diagram of the complexification of g, known as the double Vogan diagrams. They lead to a combinatorial classification of the pseudo-Hermitian symmetric spaces of G.

Affine Vogan Diagrams and Symmetric Pairs

“…The works of [16][4] are done by algebraic computations of g σ , and we intend to provide an alternative combinatorial classification by diagrams. This has been done for the pseudo-Hermitian symmetric pairs [7], and here we do the same for the remaining symplectic symmetric pairs. In Figures 3 and 4, we adopt Cartan's notation and denote the exceptional real forms by their characters, as explained in (3.12).…”

“…Figures 3 and 4 are consistent with Bieliavsky's classification [4, p.268-269], except for a minor error indicated in Remark 6.4. Together with the pseudo-Hermitian ones in [7,Figs.4,5], we obtain an independent diagrammatic classification of symplectic symmetric pairs. The classical symplectic symmetric pairs can often be studied by matrix computations on the center of g σ (see Definition 6.3).…”