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Products of Geck-Rouquier conjugacy classes and the Hecke algebra of composed permutations

We show the q-analog of a well-known result of Farahat and Higman: in the center of the Iwahori-Hecke algebra $H_{n,q}$, if $(a_{λμ}^ν(n,q))_ν$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $Γ_{λ,n}$ and $Γ_{μ,n}$, then each coefficient $a_{λμ}^ν(n,q)$ depends on $n$ and $q$ in a polynomial way. Our proof relies on the construction of a projective limit of the Hecke algebras; this projective limit is inspired by the Ivanov-Kerov algebra of partial permutations.