For linear algebraic groups $G$, the picard group is the group of characters on $G$ which is finitely generated (and will be trivial for semi-simple groups). So one does not talk of moduli space of degree zero line bundles for them. If you are talking of group varieties that are neither affine nor projective then Shafarevich's theorem states that they have unique maximal normal closed linear subgroups making the quotient an abelian variety. So it will reduce to studying on abelian varieties.

For linear algebraic groups $G$, the picard group is the group of characters on $G$ which is finitely generated (and will be trivial for semi-simple groups). So one does not talk of moduli space of degree zero line bundles for them. If you are talking of group varieties that are neither affine nor projective then Chevalley's theorem states that they have unique maximal normal closed linear subgroups making the quotient an abelian variety. So it will reduce to studying on abelian varieties.