Completed representation ring spectra of nilpotent groups

TL;DR

This paper explores the derived completion of representation rings of nilpotent groups, revealing higher homotopy structures and introducing a deformation functor that clarifies their origin, with explicit computations for p-adic Heisenberg groups.

Contribution

It introduces a deformation representation ring functor and demonstrates the equivalence of completed representation rings for finitely generated nilpotent groups.

Findings

Derived completion yields spectra with higher homotopy information.

The deformation functor R[-] explains the origin of higher homotopy classes.

Explicit computation for the p-adic Heisenberg group.

Abstract

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.