Gelfand Pairs Obtained from the Finite Upper Half Planes over \( \mathbb{Z}/2^n\mathbb{Z} \)

The general linear group \( G \) over \( \mathbb{Z}/2^n\mathbb{Z} \) acts transitively on the finite upper half plane over \( \mathbb{Z}/2^n\mathbb{Z} \), where \( \mathbb{Z} \) denotes the ring of rational integers. In this paper, it is shown that the pair of \( G \) and the stabilizer of a point on the plane is a Gelfand pair.