Many features of locally compact abelian groups carry over to commutative hypergroups, but there are some shortcomings which are not understood completely up to now. For instance, there exist many commutative hypergroups \(K\) for which the support \(\text{supp} \pi\) of the Plancherel measure is a proper subset of the dual \(\widehat K\). It is therefore interesting to have handsome criteria for \(\alpha\in \text{supp} \pi\) for particular characters \(\alpha\). Extending some results of the reviewer [Arch. Math. 56, 380-385 (1991; Zbl 0751.43005)] which are based on R. Godement’s work on positive definite functions, the authors of this paper show that \(\alpha \in\text{supp} \pi\) is equivalent to some new kind of Reiter’s condition (P2) for this specific character \(\alpha\). The authors then restrict their attention to polynomial hypergroups and derive some sufficient criteria for \(\alpha\in \text{supp} \pi\) beyond the well-known Nevai class \(M(a,b)\).