Prehomogeneous Spaces Associated with Nilpotent Orbits in Simple Real Lie Algebras E₆₍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 E₆₍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 [Jackson and Noël 05a] and [Jackson and Noël 06]. For complex or inner-type real exceptional algebras we have given general algorithms and tables in [Jackson and Noël 05b] and [Jackson and Noël 05c]. The present paper considers the case of real exceptional algebras that are not of inner type.

Keywords:

• Primary 17B05
• Secondary 17B10, 17B20, 22E30
• Nilpotent orbits
• prehomogeneous spaces