Abstract. We consider some infinitesimally unitarizable U_q(so(1,2)) highest weight modules and provide realizations of them on L²(Z/(N −1)Z), the finite dimensional Hilbert space of square integrable functions on the finite circle. Actions of the U_q(so(1,2)) generators on basis functions are given. In particular the Cartan generator is given as an explicit function of the adjacency operator. We also show how these representations at roots of unity go over into discrete series representations of so(1,2) as N goes to infinity and q goes to one. We conclude with an application of these results to the Casimir effect for a massless scalar field on the two dimensional Einstein universe, and make some comments about possible generalizations of these results to higher dimensions.

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