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Let $G$ be a compact abelian group and $\hat G={\rm hom_{\rm continuous}}(G,\mathbb{C}^*)$. Let $\mu$ be the Haar measure on $G$. For $\chi\in\hat G$, show that

$$\int_G\chi\cdot\mu=1\quad\text{if}\quad\chi=\mathbb{1}$$ $$\int_G\chi\cdot\mu=0\quad\text{if}\quad\chi\neq\mathbb{1}$$

The first one is easy and I only need to show the second. My attempt was to use the invariance properties of the Haar measure. First I tried

$$\int_G\chi(g)\cdot\mu(g)=\int_G\chi(g^{-1})\cdot\mu(g)=\int_G(\chi(g))^{-1}\cdot\mu(g)=\int_G\overline{\chi(g)}\cdot\mu(g)$$

But this only gives $\int_G\chi\cdot\mu\in\mathbb{R}$. Then I thought, if there exists a $g_0\in g$ such that $\chi(g_0)=-1$, then

$$\int_G\chi(g)\cdot\mu(g)=\int_G\chi(g_0g)\cdot\mu(g)=\int_G-\chi(g)\cdot\mu(g)\implies\int_G\chi\cdot\mu=0$$

Questions:

(1) Am I on the right track here? If so, how to prove such $g_0$ exists?

(2) This seems to be a problem of the representation theory, but it appeared in an algebraic number theory course while introducing the $L$ function. I am not sure how they are connected. Can anyone provide some backgroud?

(3) I haven't systematically studied the representation theory yet so I am not familiar with some concepts. For example, what is $\hat G$ called and what are the elements of $\hat G$ called? My teacher says elements in $\hat G$ are called characters (or characteristics?), but it doesn't seem to match the definition of a character in any of book on representation theory.

(4) Is there any text on this topic?

Appreciate any help. Thanks in advance.

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    $\begingroup$ Hint : what's so special about $-1$ ? $\endgroup$ – Maxime Ramzi May 26 '19 at 10:12
  • $\begingroup$ Yes, these continuous homomorphisms from $G$ to the unit circle in the complex plane are called "characters". (Recall: $G$ is abelian.) $\endgroup$ – GEdgar May 26 '19 at 10:18
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$\widehat{G}$ is called the dual group of $G$.

Take $g' \in G$ such that $\chi(g')\neq 1$.

$$\chi(g')\cdot \int_G \chi(g) d\mu = \int_G \chi(g'\cdot g)d\mu = \int_G \chi(g)d\mu$$

because multiplication by $\chi(g')$ simply permutes the elements of $G$.

because of the translation invariance of the Haar measure. Therefore

$$(\chi(g') - 1)\int_G \chi(g)d\mu = 0$$

and the result follows.

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    $\begingroup$ "simply permutes the elements of $G$" ... it is great to say that for a finite group. For an infinite compact group we need more: The invariance properties of Haar measure. $\endgroup$ – GEdgar May 26 '19 at 10:18
  • $\begingroup$ You're right, I've fixed it. $\endgroup$ – Lukas May 26 '19 at 10:21
  • $\begingroup$ So what are the elements of $\hat G$ called? $\endgroup$ – trisct May 26 '19 at 10:22
  • $\begingroup$ They are called characters. $\endgroup$ – Lukas May 26 '19 at 10:23
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A note on terminology: In the present context $\hat G$ is the dual group of $G$ and the elements of $\hat G$ are the characters of $G$ (not "characteristics", as far as I know). Yes, this is just one of at least two standard uses of the word "character".

Is this all part of representation theory? Yes and no:

Yes: Well, yes.

No: Usually one talks about the "representation theory" of a non-abelian (Lie) group. This is because the irreducible representations of a locally compact abelian group are one-dimensional, making various things one studies in representation theory trivial. The representation theory of (locally compact) abelian groups is often called "harmonic analysis" or "Fourier analysis".

An excellent reference is Rudin Fourier Analysis on Groups (which could have been called "Representation Theory on Locally Compact Abelian Groups".)

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