  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.
