Fixed point spaces in actions of classical algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p greater than or equal to 0, and let H be a maximal closed non-subspace subgroup of G. Given such a pair (G, H), we obtain a close to best possible upper bound for the ratio dim(x(G) boolean AND H)/dim x(G), where x is an element of G is a semisimple or unipotent element of prime order. We apply this result to the study of fixed point spaces.