Harmonic analysis on the finite symmetric space GL(n, K)GL(1, K) × GL(n − 1, K)

AbstractLet Gn = GL(n, K) and Hn = GL(1, K) × GL(n − 1, K) with K a finite field of odd characteristic. We determine all irreducible Hn-spherical representations of Gn and compute the associated spherical functions. Main by-products are: - (Gn, Hn) is not a Gelfand pair: there is one special representation which occurs twice in the regular representation of Gn on X = GnHn;- there are no irreducible spherical cuspidal representations for n>2