Special prehomogeneous vector spaces associated to F4, E6, E7, E8 and simple Jordan algebras of rank 3

Abstract

A simple complex Lie algebra has a natural grading g=g−2⊕g−1⊕g0⊕g1⊕g2 associated to the highest root. It is related to the quaternionic real form of g. The real rank of the associated symmetric space is shown to be at most 4. When the rank is equal to 4 (i.e., g≠An,Cn,G2), the semi-simple Lie algebra m=[g0,g0] is shown to be the conformal algebra of a rank 3 semi-simple Jordan algebra V. If g is one of the four exceptional Lie algebras F4,E6,E7,E8, then V is simple. The vector space g1 is a prehomogeneous vector space under the action of M×C∗, where M=Int(m) is the adjoint group of m. It admits a non-zero M-invariant polynomial of degree 4. Conversely, to any simple Jordan algebra V of rank 3 is naturally associated a representation of (a twofold covering of) its conformal group Co(V) on a vector space W, such that W is a prehomogeneous vector space under the action of Co(V)×C∗. An invariant polynomial of degree 4 is explicitly constructed. A geometric description of all the orbits of Co(V) in W is given.