Prehomogeneous Spaces Associated with Nilpotent Orbits in Simple Real Lie Algebras $\E_{6(6)}$ and $\E_{6(-26)}$ and Their Relative Invariants

Abstract

We give an efficient and stable algorithm for computing highest weights in a large class of prehomogeneous spaces associated with the nilpotent orbits of the real Lie algebras $\E_{6(6)}$ and $\E_{6(-26)}$. This paper concludes our classification of such prehomogeneous spaces for all complex and real reductive Lie algebras. For classical algebras using the fact that the nilpotent orbits are parameterized by partitions of integers we have given general formulas in Steven Glenn Jackson and Alfred G. Noël, “Prehomogeneous Spaces Associated with Complex Nilpotent Orbits,” and “Prehomogeneous Spaces Associated with Real Nilpotent Orbits.” For complex or inner-type real exceptional algebras we have given general algorithms and tables in Steven Glenn Jackson and Alfred G. Noël, “A LiE Subroutine for Computing Prehomogeneous Spaces Associated with Complex Nilpotent Orbits,” and “A LiE Subroutine for Computing Prehomogeneous Spaces Associated with Real Nilpotent Orbits.” The present paper considers the case of real exceptional algebras that are not of inner type.