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

Abstract

We investigate the structure constants of the center ${\mathcal{H}}_n$ of the group algebra $S{p}_n(q)$ over a finite field. The reflection length on the group $G{L}_{2n}(q)$ 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 $S{p}_m(q)\subset S{p}_{m+l}(q)$ and we show that the index of the centralizer of $g\in S{p}_m(q)$ in the centralizer of $g\in S{p}_{m+k}$ is equal to $q^{2ld}|S{p}_{r+l}(q)||S{p}_r(q){|}^{-1}$ for some $d$ and $r$ which are uniquely determined by the conjugacy class of $g$ in $G{L}_{2n}(q).$

46 pages

References in corpus (2)

• Stability of the centers of group algebras of $G{L}_n(q)$
• A Frobenius formula for the structure coefficients of double-class algebras of Gelfand pairs