Let $G$ be a connected algebraic group. Is it true that every $G$bundle on ${\mathbb A}^n$ is trivial? What is the reference? I am actually only interested in the case $n=2$.
This is true for $G=GL(r)$, as shown by Quillen and Suslin. For arbitrary $G$ there are counterexamples. Quite surprisingly, even $G$bundles over $\textrm{Spec }k$ may not be trivial. See the paper M. S. Raghunathan "Principal bundles on affine space and bundles on the projective line", Mathematische Annalen Volume 285, Number 2, 309332 


Over an algebraically closed field, for $G$ connected and reductive, every principal $G$bundle on ${\Bbb A}^n$ is trivial, also by a theorem of Raghunathan: "Principal bundles on affine space", in C. P. Ramanujam—a tribute, pp. 187–206, Tata Inst. Fund. Res. Studies in Math. 8 (1978). (Unfortunately I can't find this reference free online.) 


In characteristic $p$ you can make an easy counterexample with $n=1$, right? An exact sequence of commutative algebraic groups $0\to E\to X\to \mathbb A\to 0$ with $E$ an elliptic curve. 


$G$
is to be an affine algebraic group. The reductive case has been most studied, so it's also important to indicate if more generality is wanted. – Jim Humphreys Nov 15 '10 at 18:39