I want to calculate the algebraic fundamental group of a an algebraic group over a riemann surface over $\mathbb C$ (or a smooth algebraic projective curve). Let me state the first case where $\mathcal G=G\times X$ where $G$ is a linear algebraic group (affine) and $X$ is the curve. Using the Kenneth formula, it suffishs to calculate both $\pi\_1(X)$ and $\pi\_1(G)$. 


Question: For an affine algebraic group $G$,how could one calculate $\pi\_1(G)$? in particular what are $\pi\_1(GL\_r)$, $SL\_r$? and $Sp(r)$? 


Thanks in advance!



Etale Fundamental group of an algebraic group
Over $\mathbb{C}$, the étale fundamental group is the profinite completion of the topological fundamental group. You certainly assume that your affine algebraic group $G$ is connected. The question reduces easily to the case when $G$ is reductive. For $G$ reductive, the topological fundamental group of $G$ is computed in [M. Borovoi, Abelian Galois cohomology of reductive groups, Memoirs of the AMS 132 (1998), No. 626](http://www.math.tau.ac.il/~borovoi/papers/galofile.pdf). See Section 1, especially Definition 1.3 and Proposition 1.11.


