Stability of the centers of group algebras of GL _n (q)

The center Z _n (q) of the integral group algebra of the general linear group GL _n (q) over a finite field admits a filtration with respect to the reflection length. We show that the structure constants of the associated graded algebras G _n (q) are independent of n, and this stability leads to a universal stable center with positive integer structure constants which governs the algebras G _n (q) for all n. Various structure constants of the stable center are computed and several conjectures are formulated. Analogous stability properties for symmetric groups and wreath products were established earlier by Farahat-Higman and the second author.