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We investigate the structure constants of the center $\mathcal{H}_n$ of thegroup algebra $Sp_{n}(q)$ over a finite field. The reflection length on thegroup $GL_{2n}(q)$ induces a filtration on the algebras $\mathcal{H}_n$. Weprove that the structure constants of the associated filtered algebra$\mathcal{S}_n$ are independent of $n$. As a technical tool in the proof, wedetermine the growth of the centralizers under the embedding $Sp_m(q)\subsetSp_{m+l}(q)$ and we show that the index of the centralizer of $g\in Sp_m(q)$ inthe centralizer of $g\in Sp_{m+k}$ is equal to$q^{2ld}|Sp_{r+l}(q)||Sp_{r}(q)|^{-1}$ for some $d$ and $r$ which are uniquelydetermined by the conjugacy class of $g$ in $GL_{2n}(q).$

Stability of the centers of the symplectic groups rings $\matbb{Z}[Sp_{2n}(q)]$