Suppose that one has two algebraic varieties with action of a reductive group $G$: say, $X$ and $Y$.
There is an evident Kunneth-type morphism
$K_G(X) \otimes K_G(Y) \to K_G(X \times Y)$,
where the tensor product is over $R(G)$.
If $X$ is proper, and the diagonal in $X \times X$ is decomposable, it is an isomorphism.
Suppose now that $X$ is not proper, but has a decomposable diagonal. Suppose also that $Y$ is proper.
Is the morphism above still surjective/injective?
Many thanks!
