Haagerup property for quantum reflection groups

In this paper we prove that the duals of the quantum reflection groups have the Haagerup property for all $N\ge4$ and $s\in[1,\infty)$. We use the canonical arrow onto the quantum permutation groups, and we describe how the characters of the quantum reflection groups behave with respect to this canonical morphism thanks to the description of the fusion rules binding irreducible corepresentations of $H_N^{s+}$. This allows us to construct states on the central algebra of $H_{N}^{s+}$ and to use a fundamental theorem proved by M.Brannan giving a method to construct nets of trace-preserving, normal, unital and completely positive maps on the von Neumann algebra of a compact quantum group of Kac type.