We lift any (infinitesimal) unitary irreducible representation of $GL_n(\mathbb{ R})$ to a family of representations that strongly contracts to a certain type of (infinitesimal) unitary irreducible representations of $\mathbb{ R}^n\rtimes { M}_n$, with $M_n$ being the mirabolic subgroup of $GL_n(\mathbb{ R})$. For the case of $n=2$ we obtain the full unitary dual of $\mathbb{ R}^2\rtimes { M}_2$ as a strong contraction. We demonstrate the role of the Kirillov conjecture and Kirillov model for these contractions.
Categories: math-ph math.MP math.RT