Strong contraction, the mirabolic group and the Kirillov conjecture

Abstract

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.