Attached to any reductive Lie group is a “Cartan motion group” —a Lie group with the same dimension as , but a simpler group structure. A natural one-to-one correspondence between the irreducible tempered representations of and the unitary irreducible representations of , whose existence was suggested by Mackey in the 1970s, has recently been described by the author. In the present article, we use the existence of a family of groups interpolating between and to realize the bijection as a deformation: for every irreducible tempered representation of G, we build, in an appropriate Fréchet space, a family of subspaces, and evolution operators that contract onto the corresponding representation of .
"On the analogy between real reductive groups and Cartan motion groups: Contraction of irreducible tempered representations." Duke Math. J. 169 (5) 897 - 960, 1 April 2020. https://doi.org/10.1215/00127094-2019-0071