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 K^* commuting with the G-action. We show that X is determined by the -variety of fixed points under a maximal unipotent subgroup of G. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient. If G is of type A_n (n \ge 2), C_n, E_6, E_7 or 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≥5, every smooth affine SL_n -variety of dimension < 2n−2 is an SL_n -vector bundle over the smooth quotient X//SLn, with fiber isomorphic to the natural representation or its dual.