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I have recently been thnking some about algebraic groups and reading parts of Humphreys book on them, and I was wondering if there is a general process to showing they are simply connected. In particular I was wondering over other fields than $\mathbb{C}$ but if the answer only works in $\mathbb{C}$ I will settle for that.

One idea I had was that using Borel-Weil-Bott one could make a slick arguement for when the fundamental group is trivial. I would like to get away from the ad-hoc thinking process I am using.

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I apologize for typos, this was composed on my phone. – BBischof Aug 13 '10 at 5:56
Thinking spelling wrong. – anonymous Aug 13 '10 at 9:30
Can you elaborate on your Borel-Weil-Bott idea? – Sam Lichtenstein Aug 25 '10 at 4:01
BWB relates line bundles the sheaf cohomology, and thus reduce our problem to looking at invertible sheaves. It was a rough idea. – BBischof Aug 25 '10 at 4:24

Please see

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I have seen that question before, and it is relevant, but Dr. Humphreys answer here is more particularly geared at the base change(what the op wanted). I am sti holding out for a nice way to check this, but it mght follow obviously from his answer. If it does, please make it more explicit, because I am missing how. (I am a total beginner at algebraic groups). Howeverthanks for pointing this out, I should have linked it in the first place +1. – BBischof Aug 13 '10 at 13:37

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