Completed representation ring spectra of nilpotent groups

Abstract

In this paper, we examine the “derived completion” of the representation ring of a pro-$p$ group ${\mathcal{G}}_p^{\wedge }$ with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg–MacLane spectrum $ℍ\mathbb{Z}$, and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we define a deformation representation ring functor $R\left[-\right]$ from groups to ring spectra, and show that the map $R\left[{\mathcal{G}}_p^{\wedge }\right]\to R\left[\mathcal{G}\right]$ becomes an equivalence after completion when $\mathcal{G}$ is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the $p$–adic Heisenberg group.