Induced representations and harmonic analysis on finite groups

The aim of the present paper is to develop a theory of spherical functions for noncommutative Hecke algebras on finite groups. Let G be a finite group, K a subgroup and (θ, V) an irreducible, unitary K-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal basis in the commutant of IndKGV, and an associated Fourier transform. Then we translate our results in the corresponding Hecke algebra, an isomorphic algebra in the group algebra of G. Again a complete Fourier analysis is developed. As particular cases, we obtain some classical results of Curtis and Fossum on the irreducible characters. Finally, we develop a theory of Gelfand–Tsetlin bases for Hecke algebras.