Abstract
Let G be a simple simply connected algebraic group and V an irreducible module on which G acts. We prove that the variety of vectors in V which are eigenvectors for some non-central semisimple element of G has dimension strictly less than the dimension of V with a small list of possible exceptions. We also make a contribution towards the study of regular orbits in such an action; in particular, we restrict ourselves to semisimple elements. We show that provided $\dim V > \dim G$, the dimension of the union of fixed point spaces of non-central semisimple elements is less than the dimension of V with a small list of possible exceptions. In cases where the dimensions are the same we show that the complement of the union has the same dimension.