Representations on KU-modules

Algebraic topologists talk about an elevator from characteristic zero to characteristic p, with infinitely many floors in between called chromatic levels. I think you could "do representation theory" at any of these levels. I once tried to explore the first level: A "KU-module", or anyway a p-adically complete KU-module, is a level one version of a complex vector space (at level zero) and of a Zp-module (at level infinity). KU is a name for topological K-theory, in the talk I'll discuss representations --- probably, mostly, of finite groups --- on p-complete KU-modules. Representations are harder to model in this world but some computations are easier, for about the same reason that K-theory is sometimes easier to compute than cohomology.