Abstract (EN)

Attached to any reductive Lie group G is a "Cartan motion group" G0 − 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 G0, 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 G0 to realize the bijection as a deformation: for every irreducible tempered representation π of G, we build, in an appropriate Fr\'echet space, a family of subspaces and evolution operators that contract π onto the corresponding representation of G0.