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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 ringspectrum 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, andpropose to study them by analogy with the classical theory of modularrepresentations of $G$.

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