Automorphic representation of GL(1)

Question

These might be very silly questions, but somehow I am not able to understand it or I might have misunderstood something.

I am reading automorphic forms from this book.

What I have understood till now:

Automorphic forms on ${\mathrm{GL}}(1,\mathbb{A}_\mathbb{Q})$ are eventually the Dirichlet characters. In other words, if $\phi$ is an automorphic form on ${\mathrm{GL}}(1,\mathbb{A}_\mathbb{Q})$, then $\phi(g)=c\cdot \chi_{\text{idelic}}(g)\cdot |g|^{it}$ where $c \in \mathbb{C},t\in \mathbb{R}$ and $\chi_{\text{idelic}}$ is the idelic lift of a Dirichlet character $\chi$.

Now if I fix a Hecke character $\omega$, then an automorphic representation of ${\mathrm{GL}}(1,\mathbb{A}_\mathbb{Q})$ is by definiton a homomorphism $\pi: {\mathrm{GL}}(1,\mathbb{A}_\mathbb{Q}) \to {\mathrm{GL}}(V_\omega)$ where $V_\omega$ is the space of automorphic forms $\phi$ with character $\omega$. Also, we know that $V_\omega$ is one dimensional.

Now, Hecke characters of ${\mathrm{GL}}(1,\mathbb{A}_\mathbb{Q})$ are itself automorphic representations of ${\mathrm{GL}}(1,\mathbb{A}_\mathbb{Q})$.

Question 1: Are they all? If yes, then why?

We also know that there are hecke characters (See this) which does not come from Dirichlet characters. Call such a character $\psi$. Then, by definition $\psi: {\mathrm{GL}}(1,\mathbb{A}_\mathbb{Q}) \to {\mathrm{GL}}(V_\omega)$ for some Hecke character $\omega$. Also, the elements of $V_\omega$ are associated with some Dirichlet character i.e. if $\phi$ is a basis element of $V_\omega$, then $\phi(g)=c\cdot \chi_{\text{idelic}}(g)\cdot |g|^{it} $ as discussed above.

Question 2: What is the relation between $\psi$ and $\chi_{\text{idelic}}$ or $\omega$.

Please help me clear this confusion. Any suggestion regarding this would be really helpful. Thanks in advance.

Answer 1

Using your definition of automorphic representation, the answer to Question 1 is yes, tautologically. If $V_\omega$ is the space of automorphic forms "with character $\omega$" then by definition the action of the ideles on $V_\omega$ is via multiplication by the Hecke character $\omega$. In other words, the map $\mathbb{A}_{\mathbb{Q}}^\times\to \mathrm{GL}(V_\omega)=\mathbb{C}^\times$ is the Hecke character $\omega$ by definition of $V_\omega$. The book by Goldfeld and Hundley which you are following remarks after Definition 2.1.4 that their definitions are redundant in an attempt to keep the exposition of automorphic representations for $\mathrm{GL}_1$ running parallel to the exposition for $\mathrm{GL}_2$.

Similarly, in Question 2 the relationship between $\psi$ and $\omega$ is that $\psi=\omega$. The relationship between $\psi$ and $\chi_{\mathrm{idelic}}$ is that $\psi(g) = \omega(g) = \chi_{\mathrm{idelic}}(g) |g|^{it}$. We can see this last identity from $$\omega(g)\phi(x) = \phi(xg) = c\chi_{\mathrm{idelic}}(xg) |xg|^{it} = c\chi_{\mathrm{idelic}}(x)\chi_{\mathrm{idelic}}(g) |x|^{it} |g|^{it} = \chi_{\mathrm{idelic}}(g) |g|^{it} \phi(x). $$

Given the information in your question, the answers to your questions are tautological, but there is interesting mathematics baked into the background. For instance, it is an interesting property of $\mathbb{Q}$ that any (unitary) Hecke character for $\mathbb{A}_{\mathbb{Q}}$ factors as $\chi_{\mathrm{idelic}}|\cdot|^{it}$ for some Dirichlet character $\chi$ and real number $t$. When $\mathbb{Q}$ is replaced by other number fields such a decomposition may not exist.

If you continue on with the book, you will see that for $\mathrm{GL}_2$ it is important to distinguish automorphic forms from automorphic representations; roughly the idea is that irreducible automorphic representations can be infinite-dimensional vector spaces, but will always contain a unique automorphic form up to rescaling. It happens that for $\mathrm{GL}_1$ the irreducible automorphic representations are one dimensional, so this fact does not seem so interesting. If you aren't interested in $\mathrm{GL}_2$, then you might consider using a different reference for $\mathrm{GL}_1$ which does not focus on automorphic forms.