The Fourier Algebra of Certain Compact Orbit Hypergroups

Let $R$ be a compact discrete valuation ring and $G = R^d \rtimes \operatorname{GL}_d(R)$. We show that the central Fourier algebra $\operatorname{ZA}(G)$ is not amenable, reaffirming a conjecture of Alaghmandan and Spronk. Our methods reduce to studying the Fourier algebra of the commutative orbit