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

TL;DR

This paper explores how finite groups act on modules over p-adic K-theory spectra, drawing analogies with classical modular representation theory to deepen understanding of these actions.

Contribution

It introduces a novel perspective on G-actions on K-theory modules, connecting topological and algebraic representation theories.

Findings

Establishes a framework for G-actions on K-theory modules.

Draws parallels between topological and classical modular representations.

Provides new insights into the structure of G-equivariant K-theory modules.

Abstract

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 $G$.