Symmetric association schemes attached to finite upper half planes over rings

The finite upper half planes over finite fields and rings are finite analogues of the Poincaré upper half plane. The general linear group G acts transitively on the finite upper half plane. Let K denote the stabilizer of a point. In the case of fields, it is well-known that the pair of (G,K) is a Gelfand pair. In this paper, we show that (G,K) is also a Gelfand pair in the case of rings.