Let $\def\C{\mathbb{C}}T = (\C^*)^n$. A character of $T$ is defined to be a homomorphism from $T$ to $\C^*$. The characters of $T$ is of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some $a_1,\ldots, a_n \in \mathbb{Z}$.
I think we can also consider the homomorphisms from a unipotent group $U$ to $\C^*$. Are there some references which study these homomorphisms? For example, describe the expressions of these homomorphisms like in the case of of torus $T$. Thank you very much.
Suppose that $U$ is a unipotent linear algebraic group over $\mathbb{C}$, and that $f : U \to \mathbb{C}^*$ is a homomorphism of complex Lie groups. Since $\mathbb{C}^*$ is abelian, $f$ factors through the abelianisation $V = U/U'$ of $U$. Since the commutator subgroup $U'$ of $U$ is algebraic and the quotient map $p: U \to V$ is a homomorphism of algebraic groups, $V$ is also unipotent, by the preservation of Jordan decomposition under such homomorphisms. Therefore, $V$ is isomorphic to the additive group of a finite-dimensional vector space over $\mathbb{C}$. All holomorphic homomorphisms $\chi : V \to \mathbb{C}^*$ are of the form $\chi(z) = e^{\lambda(z)}$ for some linear function $\lambda \in V^*$.