Question

Here is one strategy that has not been suggested: the braid group is the fundamental group of the space $U/W$, where $U$ is the set of points in the (complexified) reflection representation of $W=S_n$ that have trivial fixer. Now $W=GL_n(\mathbb{F}_q)$ is a reflection group over the finite field $\mathbb{F}_q$; let $U$ be the set of points (over an algebraic closure $F$ of $\mathbb{F}_q$) in the reflection representation of $W$ that have trivial fixer and define the "braid group" to be the etale fundamental group of $U/W$. Essentially by definition it has a surjection onto $W$. I guess you'd want It's not clear to know whether there was a reasonable theory me in what sense this q-braid group might converge to the usual one as q goes to 1, but it should definitely play an important role in the study of reduced words for $W$ compatible with GL_n(\mathbb{F}_q)$. For instance, one might be able to define a Hecke algebra deforming $F[GL_n(\mathbb{F}_q)]$ via monodromy representations of this braid group. It's not really my thing, and I don't know the answer. A brief google/mathscinet search doesn't turn anything up.