Operator algebra K-theory has a number well-known applications to topology and geometry (for example to the Novikov conjecture in manifold theory). These stem from the index theory of Dirac operators and the Baum-Connes conjecture, but the same techniques also resonate in various ways with Lie groups and representation theory. I shall try to indicate how this comes about, focusing on one fairly new development that reconsiders an old proposal of Mackey involving the "contraction" of a Lie group to its maximal compact subgroup.