Simply connectedness of algebraic group $G$ 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 $\text{char }k=0$.
 A: 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.
