Strong contraction, the mirabolic group and the Kirillov conjecture

Abstract

We lift any (infinitesimal) unitary irreducible representation of $G{L}_n(\mathbb{R})$ to a family of representations that strongly contracts to a certain type of (infinitesimal) unitary irreducible representations of ${\mathbb{R}}^n⋊M_n$, with $M_n$ being the mirabolic subgroup of $G{L}_n(\mathbb{R})$. For the case of $n=2$ we obtain the full unitary dual of ${\mathbb{R}}^2⋊M_2$ as a strong contraction. We demonstrate the role of the Kirillov conjecture and Kirillov model for these contractions.

9 pages