Appears in collection : Representation theory and analysis on homogeneous spaces / Théorie des représentations et Analyse sur les espaces homogènes

This paper investigates a sharp analogue of Hardy's uncertainty principle for compact extensions of connected, simply connected nilpotent Lie groups, structured as semi-direct products $G=K \ltimes N$, where $K$ is a compact subgroup of automorphisms of $N$.

By leveraging Mackey's little group method, representation theory, and orbital norm structures on the dual space $\mathfrak{g}^²$, we establish exact characterizations for the joint decay of a function $f \in L^1(G) \cap L^2(G)$ and its operator-valued Fourier transform $\Theta_{\ell}(f)$.

Specifically, under Gaussian-type decay conditions governed by parameter constants $\alpha, \beta>0$ and polynomial order $m \in \mathbb{Z}_{\geq 0}$ : - If $\alpha \beta>\frac{1}{4}$, then $f=0$ almost everywhere on $G$. - If $\alpha \beta=\frac{1}{4}$, then $$ f(k, n)=f(k, t, x)=(f(k, 0, x)+H(k, t, x)) e^{-\alpha|t|^2}, $$ where $H(k, t, x)$ is a polynomial in $t$ of degree at most $2 m$ with $H(k, 0, x)=0$, and is square-integrable in $x$. - If $\alpha \beta<\frac{1}{4}$, there exist infinitely many linearly independent functions on $G$ satisfying the given decay conditions.