We investigate the structure constants of the center $\mathcal{ H}_n$ of the group algebra $Sp_{ n}(q)$ over a finite field. The reflection length on the group $GL_{ 2n}(q)$ induces a filtration on the algebras $\mathcal{ H}_n$. We prove that the structure constants of the associated filtered algebra $\mathcal{ S}_n$ are independent of $n$. As a technical tool in the proof, we determine the growth of the centralizers under the embedding $Sp_m(q)\subset Sp_{ m+l}(q)$ and we show that the index of the centralizer of $g\in Sp_m(q)$ in the 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 uniquely determined by the conjugacy class of $g$ in $GL_{ 2n}(q).$
Categories: math.RT math.CO