Stability of the centers of the symplectic group rings Z[Sp2n(q)]

Abstract

We investigate the structure constants of the center Hn of the group algebra Z[Spn(q)] over a finite field. The reflection length on the group GL2n(q) induces a filtration on the algebras Hn. We prove that the structure constants of the associated graded algebra Sn are independent of n. As tool in the proof we consider the embedding Spm↪Spn(q) and determine the behavior of the centralizers and intersection of centralizers under the embedding Spm↪Spn(q). We determine explicit formulas for rate of the growth of the centralizer of an element U∈Spm(q) in terms of dimension of the fixed space of U.