# Rational G-spectrum and geometric fixed points

For a finite group $$G$$, how is a rational $$G$$-spectrum $$X$$ detected by the geometric fixed point functor $$\phi^H$$ where we consider the conjugacy class of $$H\leq G$$? I tried finding a reference for the same but could not. The best explanation I could find so far has been Corollary 3.4.28 from Schwede's 'Global homotopy theory': For every finite group $$G$$, every orthogonal $$G$$-spectrum $$X$$ and every integer $$k$$ the map $$\left(\phi^H\circ\text{res}^G_H\right)_H:\pi^G_k(X)\longrightarrow \prod\limits_{(H)\leq G} \left(\phi^H_k(X)\right)^{W_GH}$$ becomes an isomorphism after inverting the order of $$G$$.

Is it true in the category of rational $$G$$-spectra, there is an equivalence described by the functor? $$X\longmapsto\prod\limits_{(H)\leq G} \phi^H(X)$$

Any reference and/or explanation would be appreciated. Thank you!

This is implied by Theorem 3.10 in Wimmer's "A model for genuine equivariant commutative ring spectra away from the group order". Taking $$R = \Bbb Q$$ and $$\mathcal F$$ to be the full family of subgroups of a finite group $$G$$, it says that the geometric fixed-point functors induce an equivalence $$Sp^G_{\Bbb Q} \to \prod_{[H] < G} Fun(BW_H, Sp_{\Bbb Q}).$$ So a rational genuine $$G$$-spectrum $$X$$ is equivalent data to its collection of rational spectra $$\Phi^H(X)$$ equipped with their $$W_H$$-actions.