Context: I've just started reading Tate's thesis. In it, we start with a local field $k.$ The aim of the section is to describe the structure of the character groups of $k^+$ (the additive group) and $k^*$(the multiplicative group). But for some reason when looking at the character group for $k^+$, we are looking only for the characters $\chi: k^{+} \to S^1$, where $S^1$ is the circle group but in $k^*$, we are looking at quasi characters $\chi^\prime:k^* \to \mathbb{C}^*$. Why are we doing this? @anon answered a related question, Characters of a Group: two definitions, on Math StackExchange regarding this but it really doesn't help much.

1$\begingroup$ Your link went to the question, not to an answer, so I changed it to point to the sole answer to that question. I hope that was correct. $\endgroup$– LSpiceJul 17 at 22:01

$\begingroup$ @LSpice Yes, thank you. Since there was only one answer, I didn't really think I'd need to link exactly that (and I honestly don't know how to link answers). I'll keep that in mind from next time. $\endgroup$– RitsJul 18 at 2:40

$\begingroup$ Re, every answer has a row of 'buttons' below it, the first of which is the word 'Share'. Clicking that will get a link to the answer. You can also get a link to a comment, entirely nonobviously, by clicking on the timestamp of the comment. $\endgroup$– LSpiceJul 18 at 16:05

$\begingroup$ @LSpice Ohh, thanks! $\endgroup$– RitsJul 19 at 4:29
2 Answers
I think the basic reason for the apparently different definitions boils down to the different topologies of $k^{*}$ versus $k^{+}$.
For simplicity of discussion, let's consider the case that $k= \mathbb{Q}_p$. If $\psi: \mathbb{Q}_p^{+}$ is a continuous additive character, then it must have image in the unit circle. A simple way to see this is to use that $\mathbb{Z}_p$ is compact, so its image under the continuous map $\psi$ is a compact subgroup of $\mathbb{C}^{*}$. It therefore lies in $S^1$. One can apply the same argument to the compact subgroups $p^{1} \mathbb{Z}_p$, $p^{2} \mathbb{Z}_p$, and so on.
On the other hand, one can easily define continuous quasicharacters of $\mathbb{Q}_p^*$ that do not have image in the unit circle. For instance, take $\chi(p) = p$ and $\chi(x) = 1$ for $x \in \mathbb{Z}_p^{*}$. It is still true that $\mathbb{Z}_p^*$ is a compact subgroup of $\mathbb{Q}_p^*$, but now we have $\mathbb{Q}_p^* = \bigcup_{j \in \mathbb{Z}} p^j \mathbb{Z}_p^{*}$, and the character is free to send $p$ anywhere in $\mathbb{C}^*$.

3$\begingroup$ Great explanation, and right to the point. This same idea is, in some very broad sense, at the root of why diagonalizable and unipotent groups behave so differently. $\endgroup$– LSpiceJul 17 at 21:59

3$\begingroup$ This is true for $k$ any locally compact nonarchimedean field. Any $x \in k$ is contained in a compact subgroup, such as the set of $y \in k$ with $y \leq x$. The image of any compact subgroup must be contained in $S^1$. $\endgroup$ Jul 17 at 23:00

2$\begingroup$ @KevinCasto But it is not true for the fields $\mathbb R$ and $\mathbb C$, which also appear in his work, as the identity function gives a quasicharacter that's not a character. So maybe taking circlevalued multiplicative characters is a way of forcing the fields to behave like each other, in that $\mathbb C^*$valued multiplicative characters have different natures for the archimedean and nonarchimedean fields. $\endgroup$ Jul 18 at 0:59

2$\begingroup$ @KevinCasto Oh, sorry, my comment was completely garbled. It should be characters of the additive group, where the exponential map provides the example to show the archimedean fields behave differently. $\endgroup$ Jul 18 at 17:20

1$\begingroup$ More generally, over nonarchimedean local fields, unipotent (linear) groups are ascending unions of compact subgroups. (This is what makes Jacquet modules work so well.) Not so in the archimedean case. And not for $GL(1)$ over nonarchimedean fields, either. $\endgroup$ Jul 18 at 18:23
There's a lot to absorb in Tate's thesis, but it is worth the effort. A quick, concrete answer to your question is that is allows, for $s\in\mathbb C$, functions like $n\to n^{s}$ to be a quasicharacter. This is useful for building $L$functions

2$\begingroup$ Thanks for the response. So why not even look at such quasi characters for $k^{+}$? Is it because they are just not interesting? Or maybe there is no proper structure to them, as we have for the $S^{1}$ case? Also, these quasi characters dont have a proper structure to them it seems. If we look at only the characters from $k^{*} \to $S^{1}$, can we expect them to have a nice structure? Have people tried doing it? $\endgroup$– RitsJul 17 at 19:29

2

1$\begingroup$ @LSpice haha yes, I completely failed to realize that. Even to answer my second question, there aren't any characters from $k^{*} \to \mathbb{S}^{1}$, except the trivial character, which is clear from Matt's response. $\endgroup$– RitsJul 18 at 2:39

1$\begingroup$ @Rits, well, it's not quite right to say that nonarchimedean $k$ has no unitary characters of $k^\times$: for real $t$, $x\to x_p^{it}$ on $\mathbb Q_p^\times$ is such, and these play a large role in treating Hecke characters... $\endgroup$ Jul 18 at 18:26

$\begingroup$ @paulgarrett Ohh, right. I missed that. Thanks a lot! $\endgroup$– RitsJul 19 at 4:50