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A conjecture by Mackey and Higson claims that there is close relationshipbetween irreducible representations of a real reductive group and those of itsCartan motion group. The case of irreducible tempered unitary representationshas been verified recently by Afgoustidis. We study the admissiblerepresentations of $SL(2,\mathbb{R})$ by considering families of $\D$-modulesover its flag varieties. We make a conjecture which gives a geometricunderstanding of the Makcey-Higson bijection in the general case.

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