Rational equivariant cohomology theories with toral support

Abstract

For an arbitrary compact Lie group $G$, we describe a model for rational $G$–spectra with toral geometric isotropy and show that there is a convergent Adams spectral sequence based on it. The contribution from geometric isotropy at a subgroup $K$ of the maximal torus of $G$ is captured by a module over $H^{\ast }\left(B{W}_G^e\left(K\right)\right)$ with an action of ${\pi }_0\left(W_G\left(K\right)\right)$, where $W_G^e\left(K\right)$ is the identity component of $W_G\left(K\right)=N_G\left(K\right)∕K$.