TITLE: The algebraic fundamental group of a reductive algebraic group
QUESTION [22 upvotes]: For a connected reductive algebraic group $G$ over a field $k$, other than the \'etale fundamental group of $G$ (regarded just as a scheme), there seems to be another notion, usually called the algebraic fundamental group of $G.$ I am not sure of its definition, but I guess (at least when $G$ is split) it might be something like $I/\Gamma,$ where $I$ is the "topological fundamental group" of a maximal torus $T$ (if it makes sense), and $\Gamma$ is the subgroup generated by the inverse roots. In particular, for $GL_n$ this should produce $\mathbb Z,$ which agrees with the topological fundamental group when $k=\mathbb C.$ 
Could anyone give some references on this?

REPLY [7 votes]: In addition to Marty's reference, I would recommend to look into §6 "Le groupe fondamental algébrique des groupes algébriques linéaires connexes via les resolutions flasques"
of Colliot-Thélène's paper
MR2404747, Colliot-Thélène, Jean-Louis, Résolutions flasques des groupes linéaires connexes, J. Reine Angew. Math. 618 (2008), 77–133.
This reference is in any characteristic.