Trouble understanding "orthogonal characters" and "basis of $Cl(g)$"

Question

I'm studying Georgi's book Lie Groups in Particle Physics, and I'm particularly confused by 2 statements on page 20:

1. Since the characters of different irreducible representations are orthogonal, they are different.

Isn't the character just a number, the traces, the sum of diagonal element of $D(g)$ in some representation? How can "number" be "orthogonal"?

Secondly,

2. [...]but also true that the characters are a complete basis for functions that are constant on the conjugacy classes[...]

I know each of these words mean, but I can't seem to combine them together. What does it mean to form a complete basis for a function that is constant? Am I understanding "basis for function" correctly if I say something like: The basis for $e^x$ is $1, x, x^2, \dots$ (like in Taylor's expansion)?

Thank you!

Answer 1

For each $g$, $\chi_D(g)=\operatorname{tr}D(g)$ is indeed a number. And therefore the character is a function from $G$ into $\mathbb C$. It turns out that these functions form a basis of the space of all class functions from $G$ into $\mathbb C$, like $(1,x,x^2)$ is a basis of the set of all polynomial functions with degree at most $2$ from $\mathbb C$ into itself.

Answer 2

A character (The trace of a representation) is an $L^2$ function from $G$ to $\mathbb{C}$. $L^2$ functions $G \to \mathbb{C}$ have the inner product

$$<f,h> = \int_{g \in G} f(g)\bar{h(g)} dg$$

where $dg $ is the Haar measure.

In fact, any class function of $G$ is an $L^2$ function, and since characters are class functions, they are $L^2$. Anyways, orthogonality is with respect to this inner product. Class functions form a vector space (sum of class functions is a class function, etc..), and characters form an orthogonal basis of the vector space of class functions.