Gelfand triples and their Hecke algebras. Harmonic analysis for multiplicity-free induced representations of finite groups.

A Gelfand triple is a Gelfand pair (G,K) accompanied with a representation θ of K such that the representation Ind_K^GK(θ) of G induced by θ is multiplicity-free. The main feature is the study of two concrete examples of multiplicity-free triples. The first triple in question is (GL(2,F_q),C,ν_0), where C is the Cartan subgroup of GL(2,F_q) which is isomorphic to F_{q^2} and ν_0 is an indecomposable character of F_{q^2}. This is achieved by computing explicitly the irreducible components of the induced representation Ind^{GL(2,F_q)}_C(ν_0) and the corresponding spherical functions. The task is accomplished both by application of the general theory developed in previous sections as well as by use of new ingenious methods valid in this specific case. The other triple is (GL(2,F_q),GL(2,F_{q^2}),ρ_ν), where ρ_ν is a cuspidal representation. Surprisingly, in the computation of the spherical functions corresponding to this case the decisive argument is based on the results of the previous triple.