Let X be a spectrum with an action of a group G. Given a real representation V of G, we can smash X with the one-point compactification of V to get a new G-spectrum. How does this change the equivariant homotopy type of X? Phrased this generally, we have a very classical question, and I’ll review some of the history. But I’m really interested in examples from chromatic homotopy theory, where we can use results of Devinatz-Hopkins to jump-start calculations and develop a wide-ranging formalism for describing answers. The ultimate goal is to understand the Gross-Hopkins dual of the K(n)-local sphere at low primes, but I recognize that might be a question only for the committed initiates, so I’ll structure the talk more broadly and only get to that only at the end. This with joint work with many people, but especially Hans-Werner Henn.