Say I have a reductive complex algebraic group $G$ with maximal torus $T$ and associated Weyl group $W$. I would like to be able to say that the characters of $G$ are in bijection with the $W$-invariant characters of $T$, and the bijection is given by restriction. Is this correct? If so, how could it be proven?



Describing characters of a reductive group in terms of characters of a maximal torus
I thought at first that you meant "trace character of a finite-dimensional, irreducible representation of $G$" by "character of $G$". Then this is false; for example, if $G$ is $\operatorname{SL}\_2$, then $G$ certainly has non-trivial, finite-dimensional, irreducible representations (e.g., the defining representation), but the only $W(G, T)$-fixed character of a maximal torus $T$ in $G$ is the trivial one.


If you mean literally "homomorphism $G \to \mathbb C^\times$", then certainly such a character gives a $W(G, T)$-fixed character of $T$. Conversely, a $W(G, T)$-invariant character of $T$ is trivial on $[W(G, T), T] = T \cap [G, G]$ (where the bracket notation means the group generated by commutators, not the set of commutators), and $T/[W(G, T), T] \to G/[G, G]$ is an isomorphism, so we obtain a map $G/[G, G] \to \mathbb C^\times$, as desired.


In case you're curious about that equality and isomorphism, they use the structure of reductive groups:


* Since $N\_G(T) \to W(G, T)$ is surjective, certainly $[W(G, T), T]$ is contained in $T \cap [G, G]$. Conversely, $T \cap [G, G]$ is the smallest subtorus of $T$ containing the image of $\alpha^\vee$ for every root $\alpha$ of $T$ in $G$, but $[t, w\_\alpha]$ equals $\alpha^\vee(\alpha(t))$ for every such root and every $t \in T$.
* We have just shown that $T/[W(G, T), T] \to G/[G, G]$ is an embedding. Since $G$ is the almost direct product of its maximal central torus $Z$ with $[G, G]$, already $Z \to G/[G, G]$ is surjective; but $Z$ (being a central torus) is contained in $T$, so $T/[W(G, T), G] \to G/[G, G]$ is also surjective.


Incidentally, I also used the surjectivity of $N\_G(T) \to W(G, T)$ to assert that the restriction of a homomorphism $G \to \mathbb C^\times$ to $T$ is $W(G, T)$-fixed.


