In this paper, we examine the “derived completion” of the representation ring of a pro- group 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 , 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 from groups to ring spectra, and show that the map becomes an equivalence after completion when is finitely generated nilpotent. As an application, we compute the derived completion of the representation ring of the simplest nontrivial case, the –adic Heisenberg group.
"Completed representation ring spectra of nilpotent groups." Algebr. Geom. Topol. 6 (1) 253 - 285, 2006. https://doi.org/10.2140/agt.2006.6.253