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 $G$.