The Fourier Algebra of Certain Compact Orbit Hypergroups

Let $R$ be a compact discrete valuation ring and $G=R^d⋊{\mathrm{GL}}_d(R)$. We show that the central Fourier algebra $\mathrm{ZA}(G)$ is not amenable, reaffirming a conjecture of Alaghmandan and Spronk. Our methods reduce to studying the Fourier algebra of the commutative orbit hypergroup $H=R^d\mathrm{/}{\mathrm{GL}}_d(R)$. Along the way, we also show that dual of any commutative orbit hypergroup $H$ is the hypergroup of the corresponding dual action, and this in turn shows that $\mathrm{A}(H)\cong {\mathrm{L}}^1(\widehat{H})$, which aligns with the classical setting.