A graded 𝔽p-algebra A with
action of the Steenrod algebra 𝒜∗ is said to be Steenrod presentable if there is a
polynomial ring P = 𝔽p[u1,…,un] with an action of 𝒜∗ and an 𝒜∗-invariant ideal
I ⊂ 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 φ of a simple compact Lie group G on a homogeneous Kähler
manifold X = G∕H has a Steenrod presentable equivariant cohomology for almost all
primes p if and only if φ 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.
