Universal formula for Hilbert series of minimal nilpotent orbits

Abstract

We show that the Hilbert series of the projective variety \(X=P(O_{min}),\) corresponding to the minimal nilpotent orbit \(\mathcal O_{min},\) is universal in the sense of Vogel: it is written uniformly for all simple Lie algebras in terms of Vogel's parameters \(\alpha,\beta,\gamma\) and represents a special case of the generalized hypergeometric function \({}_{4}F_{3}.\) A universal formula for the degree of \(X\) is then deduced.