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 $(T^{\ast }X{)}^{\mathrm{aff}}$ 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 $A_2$.

arXiv comment11 pages