Rational equivariant cohomology theories with toral support

For an arbitrary compact Lie group GG, we describe a model for rational GG–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 KK of the maximal torus of GG is captured by a module over H∗(BWeG(K))H∗(BWGe(K)) with an action of π0(WG(K))π0(WG(K)), where WeG(K)WGe(K) is the identity component of WG(K)=NG(K)∕KWG(K)=NG(K)∕K