Orbits and equivariant local systems combinatorics in graded Lie algebras

Abstract: In this paper, we will describe a combinatorial object to list the orbits in the ${\mathbb Z}$-graded Lie algebra, their Jordan bloc decomposition, their dimension, their dimension, the partial order and the equivariant local system (up to isomorphism) for four infinite families: two are for the symplectic groups and two are for the special orthogonal groups. These orbits and equivariant local systems appear in the study of perverse sheaves arising from graded Lie algebras.