Universal formula for the Hilbert series of minimal nilpotent orbits

We show that the Hilbert series of the projective variety X = P(Omin), corresponding to the minimal nilpotent orbit Omin, is universal in the sense of Vogel: it is written uniformly for all simple Lie algebras in terms of Vogel’s parameters α, β, γ and represents a special case of the generalized hypergeometric function 4F3. A universal formula for the degree of X is then deduced.