  This is a sequel to \cite{osy} and \cite{sxy}. Associated with $G:=\GL_n$ and
its rational representation $(\rho, M)$ over an algebraically closed filed
$\bk$, we define an enhanced algebraic group $\uG:=G\ltimes_\rho M$ which is a
product variety $\GL_n\times M$, endowed with an enhanced cross product. In
this paper, we first show that the nilpotent cone $\ucaln:=\caln(\ugg)$ of the
enhanced Lie algebra $\ugg:=\Lie(\uG)$ has finite nilpotent orbits under
adjoint $\uG$-action if and only if up to tensors with one-dimensional modules,
  $M$ is isomorphic to one of the three kinds of modules: (i) a one-dimensional
module, (ii) the natural module $\bk^n$, (iii) the linear dual of $\bk^n$ when
$n>2$; and $M$ is an irreducible module of dimension not bigger than $3$ when
$n=2$. We then investigate the geometry of enhanced nilpotent orbits when the
finiteness occurs. Our focus is on the enhanced group
$\uG=\GL(V)\ltimes_{\eta}V$ with the natural representation $(\eta, V)$ of
$\GL(V)$, for which we give a precise classification of finite nilpotent orbits
via a finite set $\scrpe$ of so-called enhanced partitions of $n=\dim V$, then
give a precise description of the closures of enhanced nilpotent orbits via
constructing so-called enhanced flag varieties. Finally, the $\uG$-equivariant
intersection cohomology decomposition on the nilpotent cone of $\ugg$ along the
closures of nilpotent orbits is established.

Representation Theory