TRANSITIVE ACTIONS AND EQUIVARIANT COHOMOLOGY AS AN UNSTABLE A*-ALGEBRA

A graded F(p)-algebra A with action of the Steenrod algebra A* is said to be Steenrod presentable if there is a polynomial ring P = F(p)[u1,..., u(n)] with an action of A* and an A*-invariant ideal I subset of P such that A = P = I and the induced action of A* on P = I is the given one. It is shown that an action phi of a simple compact Lie group G on a homogeneous Kahler manifold X = G/H has a Steenrod presentable equivariant cohomology for almost all primes p if and only if phi is conjugate to the standard action by left translation. Application to the case H = T a maximal torus reproduces a former result of the author: namely, that every topological G-action on G/T is conjugate to the standard action by left translation with isotropy group a maximal torus.