An affine variety with an action of a semisimple group G is called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group
${\mathbb {K}^{*}}$ commuting with the G-action. We show that X is determined by the
${\mathbb {K}^{*}}$-variety
$X^U$ of fixed points under a maximal unipotent subgroup
$U \subset G$. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient
$X /\!\!/ G$.
If G is of type
${\mathsf {A}_n}$ (
$n\geq 2$),
${\mathsf {C}_{n}}$,
${\mathsf {E}_{6}}$,
${\mathsf {E}_{7}}$, or
${\mathsf {E}_{8}}$, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If
$n \geq 5$, every smooth affine
$\operatorname {\mathrm {SL}}_n$-variety of dimension
$< 2n-2$ is an
$\operatorname {\mathrm {SL}}_n$-vector bundle over the smooth quotient
$X /\!\!/ \operatorname {\mathrm {SL}}_n$, with fiber isomorphic to the natural representation or its dual.