  Let $G$ be a connected reductive algebraic group over an algebraically closed
field of characteristic $p \ge 0$. We give a case-free proof of Lusztig's
conjectures [Unipotent elements in small characteristic, {\em Transform.
Groups} 10 (2005), 449--487] on so-called unipotent pieces. This presents a
uniform picture of the unipotent elements of $G$ which can be viewed as an
extension of the Dynkin--Kostant theory, but is valid without restriction on
$p$. We also obtain analogous results for the adjoint action of $G$ on its Lie
algebra $\gl$ and the coadjoint action of $G$ on $\gl^*$.
