Unitary representations of the quantum algebra su_q(2) on a real two-dimensional sphere for q in R^+ or generic q in S¹

Some time ago, Rideau and Winternitz introduced a realization of the quantum algebra su_q(2) on a real two-dimensional sphere, or a real plane, and constructed a basis for its representations in terms of q-special functions, which can be expressed in terms of q-Vilenkin functions, and are related to little q-Jacobi functions, q-spherical functions, and q-Legendre polynomials. In their study, the values of q were implicitly restricted to $q \in R^+$. In the present paper, we extend their work to the case of generic values of $q \in S^1$ (i.e., q values different from a root of unity). In addition, we unitarize the representations for both types of q values, $q \in R^+$ and generic $q \in S^1$, by determining some appropriate scalar products. From the latter, we deduce the orthonormality relations satisfied by the q-Vilenkin functions.