Let $G$ be a Lie group, $\chi:G \to \mathbb{C}$ be a class function which is a ($\mathbb{C}$-)linear combination of the irreducible characters of $G$. If $\chi=0$ on a neighborhood of $e$ in $G$, can I conclude that $\chi=0$ everywhere on $G$?
The reason I am asking this is that the fact is used in the proof of proposition (V,8.5) in Brocker and tom Dieck's 'Representation of Compact Lie Groups', with the justification '$\chi$ is analytic' given. This makes no sense to me since $G$ may not be a Riemann surface (its real dimension may not even be even), and although the book has introduced the notion of complexification $G_{\mathbb{C}}$ of a Lie group I have tried but failed to proceed using this construction, namely I cannot make sure the extended characters vanish in a neighborhood of $G_\mathbb{C}$.
Any help is appreciated!
You don't need to be on a Riemann surface or a complex manifold to have a notion of analytic function. A function $f$ on some open subset $U \subset \mathbb{R}^n$ is analytic if it has a convergent power series around every point. It's not hard to show that if $U$ is connected and $f = 0$ on some open set $V \subset U$ then $f = 0$ on all of $U$.
This is a local condition, so to do it on a manifold you just take some chart containing your point and see if a function is analytic when you view it as a function on $\mathbb{R}^n$. There is a bit of subtlety in that you need to be sure that whether a function is analytic or not at a point should not depend on the choice of chart around that point. It turns out that if you require the gluing maps defining your manifold to be analytic (rather than just $C^\infty$) this automatically works, and you have what is known as a real-analytic manifold.
Now moving back into Lie theory, this proof is alluding to a couple facts:
1) Lie groups are real-analytic manifolds. More precisely, any Lie group (a priori a smooth manifold) admits a unique real analytic structure such that the multiplication and inverse maps are real analytic functions.
2) Any finite dimensional smooth representation $G \to GL_n(\mathbb{C})$ is an analytic map relative to the above analytic structure, and in particular so are the characters.
Another minor point: if your Lie group is disconnected this theorem is false, you can just pull back any nontrivial representation of $G/G_0$.