Induced representations and harmonic analysis on finite groups

TL;DR

This paper develops a Fourier analysis framework for induced representations and Hecke algebras of finite groups, providing new bases and transforms that generalize classical harmonic analysis techniques.

Contribution

It introduces an orthogonal basis and Fourier transform for the commutant of induced representations, extending harmonic analysis to Hecke algebras and Gelfand-Tsetlin bases.

Findings

Constructed an orthogonal basis for the commutant of induced representations.

Developed a complete Fourier transform for Hecke algebras.

Extended results to Gelfand-Tsetlin bases for Hecke algebras.

Abstract

Given a finite group $G$ and a subgroup $K$, we study the commutant of $\text{Ind}_K^G\theta$, where $\theta$ is an irreducible $K$-representation. After a careful analysis of Frobenius reciprocity, we are able to introduce an orthogonal basis in such commutant 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 and, as particular cases, we obtain some results of Curtis and Fossum on the irreducible characters of Hecke algebras. Finally, we develop a theory of Gelfand-Tsetlin bases for Hecke algebras.