Fixed point spaces in primitive actions of simple algebraic groups

Let G be a simple algebraic group of adjoint type acting primitively on an algebraic variety Ω. We study the dimensions of the subvarieties of fixed points of involutions in G. In particular, we obtain a close to best possible function f(h), where h is the Coxeter number of G, with the property that with the exception of a small finite number of cases, there exists an involution t in G such that the dimension of the fixed point space of t is at least f(h)dimΩ.