Representations of finite groups on modules over K-theory (with an appendix by Akhil Mathew)

Let $G$ be a finite group, and let $\mathbf{K}_p$ denote the completion at $p$ of the complex $K$-theory spectrum. $\mathbf{K}_p$ is a commutative ring spectrum that in some ways is very similar to the usual ring $\mathbf{Z}_p$ of $p$-adic integers. We discuss $G$-actions on $\mathbf{K}_p$-modules, and propose to study them by analogy with the classical theory of modular representations of $G$.