Mackey analogy via -modules for SL(2, ℝ)

A conjecture by Mackey and Higson claims that there is close relationship between irreducible representations of a real reductive group and those of its Cartan motion group. The case of irreducible tempered unitary representations has been verified recently by Afgoustidis. We study the admissible representations of SL(2, ℝ) by considering families of 𝒟-modules over its flag varieties. We make a conjecture which gives a geometric understanding of the Makecy–Higson bijection in the general case.