Minimal Nilpotent Orbits of type G2, F4 and E8

Abstract

Let $\mathfrak g$ be a complex simple Lie algebra of type $G_2$, $F_4$, or $E_8$. We prove that the closure $\overline{\mathcal O}_{\min}(\mathfrak g)$ of its minimal nilpotent orbit is not isomorphic to $(T^*X)^{\mathrm{aff}}$ for any smooth quasi-affine variety $X$. We do not assume that the isomorphism is equivariant or Poisson. We also prove the same statement in types $A_1$ and $A_2$.