Pontrajagin-Cartier duality for Spherical group rings

Let $G$ be a compact Lie group, $\hat{G}:=Ho{m}_{cont}(G,\mathbb{T})$ its Pontrajagin dual, $R$ an $E_{\infty }$ ring spectrum.

Lemma: there’s an equivalence

$$R^{hG}\simeq R[\hat{G}]$$

Remark Geometrically we can think of as $G$-fixed points with a trivial action asglobal sections on a certain “Betti” stack

$R^{hG}\simeq {\lim }_{BG}R\simeq \Gamma \mathcal{O}(B{G}_R)$ where $B{G}_R$ is the constant functor $CAl{g}_R\to S$ landing in $BG$.

Example $\hat{\mathbb{T}}\simeq \mathbb{Z}$, hence ${\mathbb{G}}_{m,R}\simeq Spec(R^{h\mathbb{T}})$.

Remark One can imagine a version of the above, where the ring is equipped with a $\mathbb{T}$-action. Then setting $R\simeq THH(A)$ for an $E_{\infty }$-ring $A$, this then would give a certain dual interpretation of $Spec(NC(A))$ where RHS is the $E_{\infty }$ spectrum encoding negative cyclic cohomology .

Example $\widehat{C_p}\simeq {\mathbb{C}}_p$ 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 $G$ is a compact abelian lie group $Spec(R^{hG})\simeq Hom(\hat{G},Spec(R^{h\mathbb{T}}))\simeq Hom(\hat{G},{\mathbb{G}}_{m,R})$

And then the non-abelian case is left Kan-extended from the abelian case.

Cartier duality of spherical group rings at higher height