  Attached to any reductive Lie group $G$ is a "Cartan motion group" $G_0$ $-$
a Lie group with the same dimension as $G$, but a simpler group structure. A
natural one-to-one correspondence between the irreducible tempered
representations of $G$ and the unitary irreducible representations of $G_0$,
whose existence had been suggested by Mackey in the 1970s, has recently been
described by the author. In the present notes, we use the existence of a family
of groups interpolating between $G$ and $G_0$ to realize the bijection as a
deformation: for every irreducible tempered representation $\pi$ of G, we
build, in an appropriate Fr\'echet space, a family of subspaces and evolution
operators that contract $\pi$ onto the corresponding representation of $G_0$.
