Completed representation ring spectra of nilpotent groups

In this paper, we examine the “derived completion” of the representation ring of a pro-p group G∧ p 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 HZ , 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[−] from groups to ring spectra, and show that the map R[G∧ p ]→ R[G] becomes an equivalence after completion when 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.