On Irreducibility of an Induced Representation of a Simply Connected Nilpotent Lie Group

Abstract

Let $G$ be a simply connected nilpotent Lie group, $\mathcal{G}$ the finite-dimensional Lie algebra of $G$, $\mathcal{V}$ a finite-dimensional vector space over $\mathbb{C}$ or $\mathbb{R}$, and $H$ a connected Lie subgroup of $G$ such that the Lie algebra of $H$ is a subordinate subalgebra to an element $\pi $ of $Hom\left( \mathcal{G},gl\left( \mathcal{V}\right) \right) $. In this work, we construct an irreducible representation $\chi _{\pi }$ of $H$ such that the induced of $ \chi _{\pi }$ on $G$ is irreducible.