Mackey analogy via $\mathcal{D}$-modules in the example of $SL(2,\mathbb{R})$

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 $SL(2,\mathbb{R})$ by considering families of $\D$-modules over its flag varieties. We make a conjecture which gives a geometric understanding of the Makcey-Higson bijection in the general case.

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