Small G-varieties

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.