Motivic Adams isomorphism and tom Dieck splitting in characteristic zero

We prove a motivic Adams isomorphism for smooth affine linearly reductive groups over fields of characteristic zero. The diagonal of the classifying stack defines an invertible twist. We calculate the twist for finite groups and split tori, and show why it need not be a constant sphere for a split reductive group. We show that geometric fixed points identify each isotropy layer with the lisse motivic category of its Weyl group. Furthermore, over an algebraically closed field, a normalizer transfer and a isotropy filtration argument give a twisted tom Dieck splitting for split spectra, including suspension spectra.