  Let $G$ be a connected semisimple algebraic group of adjoint type defined
over an algebraically closed field $K$ of positive characteristic. The
characteristic $p$ is very good for $G$ when $p$ is suitably large and, if $G$
is of type $A_n$, $p$ does not divide $n+1$. The majority of results concerning
the geometric structure of algebraic groups in positive characteristic are
valid only in very good characteristic. We demonstrate that the dual nilcone
$\mathcal{N}^* \subseteq \mathfrak{g}^*$ is a normal variety in certain
characteristics which are not very good for $G$. As an application, we extend
the results of Ardakov and Wadsley on representations of $p$-adic Lie groups.
Under further restrictions on the characteristic, we show that the canonical
dimension of a coadmissible representation of a semisimple $p$-adic Lie group
in a $p$-adic Banach space is either zero or at least half the dimension of a
nonzero coadjoint orbit.
