  For absolutely simple, finite-dimensional Lie algebras g of rank at least 2,
defined over a local field of characteristic 0 and admitting a graduation:
g=g(-2)+g(-1)+g(0)+g(1)+g(2) given by an element H such that 2H is simple, we
construct parabolic subgroups P of the automorphism group of g which centralize
H, having geometric prehomogeneous action on g(1) and g(-1). We study the
structure of these prehomogeneous vector spaces. We prove that the Zeta
functions associated to the fundamental invariants for the P action on g(1) and
g(-1) have meromorphic extensions which satisfy functional equations. We give
the explicit calculus of the coefficients of these functional equations and the
Bernstein polynomials associated to these fundamental invariants in the
archimedian case, by reducing the problem to a similar problem for centralizers
of pair of commuting sl(2) Lie algebras. This work is a generalization of
well-known results when g(2)=0.

Representation Theory