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Induced representations and harmonic analysis on finite groups

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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.

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Authors and Affiliations

  1. Dipartimento SBAI, Università degli Studi di Roma “La Sapienza”, via A. Scarpa 8, 00161, Rome, Italy

    Fabio Scarabotti

  2. Dipartimento di Matematica e Fisica, Università Roma TRE, L. San Leonardo Murialdo 1, 00146, Rome, Italy

    Filippo Tolli

Authors
  1. Fabio Scarabotti
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  2. Filippo Tolli
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Correspondence to Filippo Tolli.

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Communicated by A. Constantin.

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Scarabotti, F., Tolli, F. Induced representations and harmonic analysis on finite groups. Monatsh Math 181, 937–965 (2016). https://doi.org/10.1007/s00605-016-0918-9

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  • DOI: https://doi.org/10.1007/s00605-016-0918-9

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