Orbits and equivariant local systems combinatorics in graded Lie algebras

TL;DR

This paper introduces a combinatorial framework to classify orbits, their decompositions, and local systems in graded Lie algebras associated with symplectic and orthogonal groups, aiding the study of perverse sheaves.

Contribution

It provides explicit combinatorial descriptions for orbits and local systems in graded Lie algebras of classical groups, expanding understanding of their structure.

Findings

Describes combinatorial objects for orbit classification

Provides formulas for Jordan block decompositions

Analyzes equivariant local systems 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.