TITLE: Characters on a torus.
QUESTION [6 upvotes]: Let $T = (\mathbb{C}^*)^n$. It is said that the characters on $T$ must be of the form $f(t_1,\ldots,t_n)=t_1^{a_1}\cdots t_n^{a_n}$ for some $a_1,\ldots, a_n \in \mathbb{Z}$. Is it possible that $a_i \in \mathbb{C}\backslash \mathbb{Z}$? Thank you very much.

REPLY [3 votes]: In general, the expression $t^a$ is multi-valued if $t \in \mathbf C^\times$ and $a \in \mathbf C$, so what you have written is not well-defined. In fact, since $\mathbf C^\times \cong S^1 \times \mathbf R^\times_{>0}$, a character of $\mathbf C^\times$ decomposes as the product of a character of $S^1$ and a character of $\mathbf R^\times_{>0}$. Characters of $S^1$ are all of the form $z \mapsto z^n$ for $n \in \mathbf Z$. On the other hand, the log gives an isomorphism $\mathbf R^\times_{>0} \cong \mathbf R$, and characters of $\mathbf R$ are all of the form $x \mapsto e^{2\pi i x t}$ for $t \in \mathbf R$. Can you piece this together?