Character and representations of $\Bbb C^\times$

Question

I wanted to find all representations of the group $\Bbb C^\times$ under multiplication. I was thinking that I could have a degree $n$-representation of this for any $n\in \Bbb N$ by letting $z\in \Bbb C^\times$ have matrix $\rho_z=zI_{n\times n}$ which acts on $\Bbb C^n$ and is a group homomorphism.

Then I was wondering if I could decompose this into subspaces and was considering using character theory, but I realised I only know character theory for finite groups and it wasn't being clear to me how to be rigorous here. The character $\chi$ seems to be $nz$ for $z\in \Bbb C^\times$ and I wanted to find if I could just decompose this into one dimensional representations of $\Bbb C^\times$ where I simply act by $[z]$ on $\Bbb C^1$, and the character of this is of course $\chi_u(z)=z$. With that in mind I would check $$(\chi|\chi_u)=\sum_{t\in G} \chi(t)\chi_u(t)^*,$$ but then this sum is not finite, and by symmetry I can see certain things, but I am uncomfortable with uncountable sums.

How to find the representations of $\Bbb C^\times$ and can I use character theory for groups somehow?

Answer 1

Representations: Irreducible representations (irreps) of abelian groups are one dimensional, so you can start by considering one dimensional representations of $\mathbb C^\times$. An irrep of $\mathbb C^\times$ is a homomorphism $\rho:\mathbb C^\times \to \mathrm{GL}(1,\mathbb C) = \mathbb C^\times$. Now let us assume that an arbitrary algebraic irrep $\rho: \mathbb C^\times \to \mathbb C^\times$ has the following Laurent expansion around $z =0$: $$ \rho(z) = \sum_{n=\infty}^\infty a_n z^n\,, \qquad z \in \mathbb C^\times\,. \tag{L}$$ Since $\rho$ is a group homomorphism, it has to satisfy: $$ \rho(wz) = \rho(w) \rho(z)\,, \tag{H}$$ for all $w, z \in \mathbb C^\times$. Substituting (L) in (H) we get an equation: $$ \sum_{n=-\infty}^\infty a_n w^n z^n = \sum_{m=-\infty}^\infty \sum_{n=-\infty}^\infty a_m a_n w^m z^n\,.$$ This equation can be satisfied for all $w, z \in \mathbb C^\times$ only if we impose the constraint that for all but one $n \in \mathbb Z$ the coefficients $a_n$ must vanish and the nonzero $a_n$ must be equal to $1$. So the conclusion is that irreps of $\mathbb C^\times$ are parametrized by integers and they are all of the form: $$\rho_n(z) = z^n\,, \qquad n \in \mathbb Z\,. \tag{I}$$ A general representation of $\mathbb C^\times$ is a direct sum of its irreps.

Characters: The irreps of $\mathbb C^\times$ are also called characters of $\mathbb C^\times$ since the definition of a multiplicative character (a homomorphism to $\mathbb C^\times$) coincides with the definition of the irreps.

This classification can be found in the proposition 1.9 of this note.

Edit: As pointed out in the comment by N.H., this calssification does not apply to all representations. In addition to being group homomorphism, a restriction of being morphism of algebraic variety is also assumed. These are called "algebraic" representations (definition 1.1 of the note). A particular example of a representation that is excluded from this type is $\rho(z) = \bar z^n$, though this is related to an algebraic or analytic representation via complex conjugation.