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$G$ is a semisimple algebraic group over $k$, if $G_{\bar k}$ is simply connected when we do base change to $\bar k$, can we descent the simply connectedness to $G$?

Here, simply connectedness means no nontrivial connected central isogeny onto $G$.

Can we say that simply connected algebraic group is geometrically connected? If then we can give an affirmative answer by considering the universal cover of $G$.

Welcome for any answer under further assumption that $\text{char }k=0$.

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  • $\begingroup$ What if we assume char=0? $\endgroup$
    – TJCM
    Commented Mar 25, 2010 at 2:54
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    $\begingroup$ If we assume char 0 then the given definition of "simply connected" is correct, and the whole thing becomes an easy exercise in Galois descent. (The magic of the theory of connected semisimple groups is that it works in an essentially characteristic-free way if one defines things properly.) I recommend that if you are only interested in char. 0 then you treat this as an exercise in Galois descent and try to solve it for yourself accordingly via that technique (using a suitable "universal property" of simply connected cover over $\overline{k}$ so as to get the Galois descent datum on it). $\endgroup$
    – BCnrd
    Commented Mar 25, 2010 at 3:09

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To amplify Brian Conrad's semi-answer, I need a more precise definition of "simply connected" at the outset. In characteristic 0 some of the classical ways of thinking about this concept can be carried over to the algebraic setting, but in prime characteristic the most common definition starts with a connected semisimple group. Over an algebraically closed field, the algebraic criterion for such a group to be simply connected is that the character group of a maximal torus be the full weight lattice.

Here the "fundamental group" of the adjoint group in the compact case is re-interpreted as the quotient of the weight lattice by the root lattice, which may also be regarded as the (scheme-theoretic) center of the simply connected group.

There may be no quotable source earlier than the 1956-58 Chevalley seminar. The classification work of Tits and others then descends to arbitrary fields of definition. In SGA 3, Expose 22 (by Demazure), Definition 4.3.3 defines "simply connected" in terms of the behavior of fibers relative to this criterion using the root datum language.

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