Mathews' Identity for Prehomogeneous Vector Spaces of Heisenberg Parabolic Type

Abstract

Some aspects of the invariant theory of a prehomogeneous vector space of Heisenberg parabolic type are studied. In particular, it is shown that a classical identity given by George Ballard Mathews for the space of binary cubic forms has a natural explanation in terms of the Bruhat decomposition associated with the parabolic subgroup and consequently admits a generalization to all prehomogeneous vector spaces of this type. The results are expected to play a role in the definition of an analogue of the Kelvin transform for certain conformally invariant systems of differential equations that have previously been associated with these spaces.

Key Words:

• Invariant theory
• Prehomogeneous vector space
• Simple algebraic group

2000 Mathematics Subject Classification:

• Primary 11S90
• Secondary 17B20

Notes

Communicated by D. Nakamo.

It is a pleasure to thank the NSA, which partially supported this work through grant H98230-07-01-0020.