A remark on homogeneous affine varieties and related matters
  In this note we give an example of affine quotient $G/H$ where $G$ is an affine algebraic group over an algebraically closed field of characteristic 0 and $H$ is a unipotent subgroup not contained in the unipotent radical of $G$. Some remarks about symmetric algebras of centralizers of nilpotent elements in simple Lie algebras, in particular cases, are added. 
Jean-Yves Charbonnel (IMJ)