  The center $\mathscr{Z}_n(q)$ 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 $\mathscr{G}_n(q)$ are independent of $n$, and this stability
leads to a universal stable center with positive integer structure constants
which governs the algebras $\mathscr{G}_n(q)$ 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.
