For any abelian variety $A$, there is a dual abelian variety $\hat{A}$ which parametrizes degree zero line bundles.


Is it possible to expect similar duality for group varieties (suppose over $\mathbb{C}$ for simplicity)? I noticed that there is a notion of Cartier duality, but I feel this is not what I am looking for. 


Besides, what is wrong if one try to look at the moduli space of degree zero line bundles on group variety just as the case in abelian variety?



Duality for group variety
The "degree" of a line bundle is not defined in general, it is better to study $\textrm{Pic}^0$, the connected component of the indentity of the Picard scheme. For abelian varieties this is exactly the dual abelian variety.


As noted in the comments, the Picard group of a linear algebraic group $G$ is finite, so $\textrm{Pic}^0(G)=0$ and hence it is not very interesting. By Chevalley's theorem, we may write any algebraic group $H$ as 
$$0 \to G \to H \to A \to 0,$$
where $G$ is linear algebraic and $A$ is abelian. I claim that $\textrm{Pic}^0(H) = \textrm{Pic}^0(A) = \widehat{A}$, so one can reduce to the case of abelian varieties.
Here is a sketch of a proof. By [1, Prop. 6.10], we have an exact sequence
$$0 \to \textrm{Pic}(A) \to \textrm{Pic}(H) \to \textrm{Pic}(G).$$
As already noted, $\textrm{Pic}(G)$ is finite. Hence when we restrict this exact sequence, to the identity component, we obtain $\textrm{Pic}^0(H) = \textrm{Pic}^0(A)$, as required.


[1] Sansuc - Groupe de Brauer et arithmétique des groupes algébriques linéaires.


