In this chapter we want to consider finite analogues of the symmetric spaces G/K mentioned above. These are related to the concept of Gelfand pair (G, K). See also Gelfand [1988], Helgason [1968, 1984], Selberg [1989], Terras [1985 and 1988], and Vilenkin [1968]. In particular, we will study a finite analogue of the Poincaré upper half plane H ≅ G/K, with G = SL(2, ℝ) and K = SO(2). We will replace ℝ with the finite field q. Such quotients have been considered by Angel et al. [1992], Celniker et al. [1993], Soto-Andrade [1987], and Terras [1991, 1996]. Other quotients G/B, for B the Borel subgroup of upper triangular matrices, have been considered by Krieg [1990] and Stanton [1990], for example.

One goal is to obtain a Fourier transform on K\G/K called the spherical transform (see Corollary 1 of Theorem 2 in Chapter 20 and formula (15) of Chapter 23). The spherical transform has many of the properties of the Fourier transform on abelian groups since it is complex valued rather than matrix valued.