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- W2597303965 abstract "Let G be a finite group, K a subgroup and (σ, V ) an irreducible representation of K. Then the Hecke algebra associated with the triple (G,K,σ) is the commutant of the induced representation IndKσ. In [3] Curtis and Fossum derived several explicit expressions for the characters of Hecke algebras. In the present paper we give an exposition of their results (see also [5], pp. 279-291) in the language of finite harmonic analysis. In particular, we show the connection with the theory of finite Gelfand pairs. 1 – Introduction Let K ≤ G be finite groups and denote by X = G/K the corresponding homogeneous space. A function f : G −→ C is said to be bi-K-invariant if f(k1gk2) = f(g) for all g ∈ G, k1, k2 ∈ K. The bi-K-invariant functions form an algebra under convolution that coincides with the commutant of the permutation representation of G on X. In other words, any G-invariant operator on X is given by the convolution with a suitable bi-K-invariant kernel. We recall that (G,K) is a (finite) Gelfand pair when the permutation representation of G on the space G/K decomposes without multiplicity; equivalently, when the algebra of bi-K-invariant functions is commutative. In this setting it is possible to develop a harmonic analysis based on a particular basis of the space of bi-K-invariant functions constitued by the so called spherical functions. The theory of spherical functions has many applications; we refer to ([6, 13, 1]) for complete" @default.
- W2597303965 created "2017-03-23" @default.
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- W2597303965 date "2013-01-01" @default.
- W2597303965 modified "2023-09-25" @default.
- W2597303965 title "Hecke algebras and harmonic analysis on finite groups" @default.
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