The smallest quantum Mackey deformation

When $G$ is a real semisimple group, there is a surprising interplay between its representation theory and that of its motion group $G_0$, known as the Mackey analogy. The present paper extends this analogy to the framework of $q$-deformations, for $G = \mathrm{SL}(2,\mathbb{R})$. In fact, we construct a deformation of $\mathrm{SL}(2,\mathbb{R})$ parametrized by $(q,t) \in \mathbb{R}_+^* \times \mathbb{R}$, where $q$ is the quantization parameter and $t$ is the Mackey parameter. We show how the representation theory varies along this deformation and we prove an analogue of the Connes-Kasparov isomorphism for the $q$-deformed reduced group C*-algebra.