Rational stable homotopy type of equivariant projective spaces and Grassmannians

We prove explicit rational stable splittings of equivariant complex projective spaces $\mathbb{C}P(V)$ and Grassmannians $Gr_n(V)$, for complex representations $V$. When $V$ is a sum of one-dimensional representations, both $\mathbb{C}P(V)$ and $Gr_n(V)$ are rationally a wedge of representation spheres. For general finite groups $G$ and $V$ a sum of irreducible representations which are not necessarily one-dimensional, we show that $\mathbb{C}P(V)$ splits rationally as a wedge of Thom spaces over irreducible factors in $V$. For $Gr_n(V)$, the factors in the corresponding rational splitting are a smash product of Thom spaces over lower Grassmannians on irreducible factors in $V$.