Induced representations and harmonic analysis on finite groups

Abstract

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 $$(\theta ,V)$$ an irreducible, unitary K-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal basis in the commutant of $$\text {Ind}_K^GV$$ , 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.