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If $G$ is an abelian group, the characters associated to the rapresentations of $G$ over $\textrm{GL}_1(\mathbb C)=\mathbb C^\ast$ are simply the group homomorphisms:

$$\chi:G\longrightarrow\mathbb C^\ast$$

On the contrary if $G$ is a topological group (assume locally compact) then a character is a continous homomorphism: $$\chi':G\longrightarrow\mathbb R/\mathbb Z\cong S^1$$

Why do we have two apparently different definitions? I know that the range of $\chi$ (first definition) is $S^1$ only when $G$ is finite.

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    $\begingroup$ I've seen the first definition referred to as quasi-characters, eg. in Tate's thesis. $\endgroup$ Commented Jul 16, 2014 at 10:26
  • $\begingroup$ Wikipedia call them simply characters: en.wikipedia.org/wiki/Character_group I'm a bit confused $\endgroup$
    – Dubious
    Commented Jul 16, 2014 at 10:32
  • $\begingroup$ Oh yeah abelian groups. I'm sorry $\endgroup$
    – Dubious
    Commented Jul 16, 2014 at 10:33

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Agreeing with Adam in the comments: if the bigger codomain $\Bbb C^*$ is used we have what number theory would term a quasicharacter, at least from what I've read. True, from the viewpoint of representation theory, it's just a one-dimensional character. However for convenience and elegance we sometimes impose unitarity on our representations, and stipulating that the codomain be $S^1$ is essentially saying we want it to be a unitary one-dimensional representation.

KCd in another thread states why $U(1)$ is so useful: it is the "universal dualizer," which is where we get Pontryagin duality. Precisely: $G\cong\hom(\hom(G,A),A)$ for all loc. comp. $G$ iff $A=U(1)$.

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