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.

8 pages