Appears in collection : 2025 - T1 - WS2 - Tempered representations and K-theory

Recently, Nigel Higson and Alexandre Afgoustidis made precise an analogy proposed by George Mackey between some unitary representations of a semisimple Lie group and unitary representations of its associated semidirect product group. In this talk, I will show a construction of an embedding of the reduced group C_-algebra of the Cartan motion group into the reduced group C_-algebra of the reductive group. This can then be used to characterize the Mackey bijection. We shall discuss the case of the complex reductive group before proceeding to discuss the difficulty behind the construction for a real reductive group.