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 $(a_{\lambda\mu}^{\nu}(n,q))_\nu$ is the set of structure constants involved in the product of two Geck-Rouquier conjugacy classes $\Gamma_{\lambda,n}$ and $\Gamma_{\mu,n}$, then each coefficient $a_{\lambda\mu}^{\nu}(n,q)$ 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.