The Fourier Algebra of Certain Compact Orbit Hypergroups

Abstract: 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 hypergroup $H = R<sup>d/\operatorname{GL}_d(R)$. 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 $$\operatorname{A}(H) \cong \operatorname{L}<sup>1(\widehat{H})$$, which aligns with the classical setting.