Let be a compact Lie group, its Pontrajagin dual, an ring spectrum.
Lemma: there’s an equivalence
Remark Geometrically we can think of as -fixed points with a trivial action asglobal sections on a certain “Betti” stack
where is the constant functor landing in .
Example , hence .
Remark One can imagine a version of the above, where the ring is equipped with a -action. Then setting for an -ring , this then would give a certain dual interpretation of where RHS is the spectrum encoding negative cyclic cohomology .
Example we get variants of the example above which can be used to give a geometric interpretation to the cyclotomic structure.
As it often happens, the general abelian case can be expressed in terms of the torus case:
Lemma if is a compact abelian lie group
And then the non-abelian case is left Kan-extended from the abelian case.