Langlands functorality in K-theory for C∗-algebras. I. Base change

We compare representations of the real and complex general linear groups and special linear groups in the framework of K-theory, using base change on L-parameters. We introduce a notion of base change on K-theory involving the fixed point set of the reduced dual of a complex group. For general linear groups, we prove that the base change map is compatible with the Connes-Kasparov isomorphism.