  A celebrated result of Farahat and Higman constructs an algebra $\mathrm{FH}$
which "interpolates" the centres $Z(\mathbb{Z}S_n)$ of group algebras of the
symmetric groups $S_n$. We extend these results from symmetric group algebras
to type $A$ Iwahori-Hecke algebras, $H_n(q)$. In particular, we explain how to
construct an algebra $\mathrm{FH}_q$ "interpolating" the centres $Z(H_n(q))$.
We prove that $\mathrm{FH}_q$ is isomorphic to $\mathcal{R}[q,q^{-1}]
\otimes_{\mathbb{Z}} \Lambda$ (where $\mathcal{R}$ is the ring of
integer-valued polynomials, and $\Lambda$ is the ring of symmetric functions).
The isomorphism can be described as "evaluation at Jucys-Murphy elements",
leading to a proof of a conjecture of Francis and Wang. This yields character
formulae for the Geck-Rouquier basis of $Z(H_n(q))$ when acting on Specht
modules.
