Describing characters of a reductive group in terms of characters of a maximal torus

Question

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?

Answer 1

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.

Answer 2

Let $G$ be a connected reductive group over an algebraically closed field $k$. Write $G^{\rm ss}=[G,G]$ (which is semisimple) and let $G^{\rm sc}$ denote the universal cover of the semisimple group $G^{\rm ss}$; then $G^{\rm sc}$ is simply connected. Consider the composite homomorphism $$\rho\colon G^{\rm sc}\twoheadrightarrow G^{\rm sc}\hookrightarrow G.$$

Let $T\subseteq G$ be a maximal torus. Consider the maximal tori $T^{\rm ss}=T\cap G^{\rm ss}\subset G^{\rm ss}$, $T^{\rm sc}=\rho^{-1}(T)\subset G^{\rm sc}$. We write $X^*$ for the character group, and $X_*$ for the cocharacter group. Clearly, \begin{multline*} X^*(G)\cong\{x\in X^*(T)\ |\ x|_{T^{\rm ss}}=1\}\\ =\{x\in X^*(T)\ |\ \langle x,u\rangle=0\ \ \forall u\in X_*(T^{\rm ss})\} =:X_*(T^{\rm ss})^\bot \end{multline*}

Proposition. $X^*(G)\cong X^*(T)^W$.

Proof. Write $X=X^*(T)$, $X^\vee=X_*(T)$. Let $R=R(G,T)\subset X$ denote the root system, and $R^\vee=R^\vee(G,T)\subset X^\vee$ denote the coroot system. Let $B\subset G$ be a Borel subgroup containing $T$. Let $S=S(G,T,B)\subset R\subset X$ denote the system of simple roots, and $S^\vee=S^\vee(G,T,B)\subset R^\vee\subset X^\vee$ denote the system of simple coroots. Then $S^\vee$ is a basis of $X_*(T^{\rm sc})$ embedded in $X$.

Since the homomorphism $T^{\rm sc}\to T^{\rm ss}$ is surjective with finite kernel, the induced homomorphism $X_*(T^{\rm sc})\to X_*(T^{\rm ss})$ is injective with finite cokernel. We conclude that $$ X^*(G)\cong X_*(T^{\rm ss})^\bot =X_*(T^{\rm sc})^\bot= (S^\vee)^\bot.$$

We know that the Weyl group $W=W(G,T)$ is generated by the reflections $s_\alpha$ for $\alpha\in S$. These reflections act on $X=X^*(T)$ by $$s_\alpha(x)= x-\langle x,\alpha^\vee\rangle \alpha\quad\ \text{for}\ \, x\in X, \,\alpha\in S;$$ see, for instance, Subsection 1.1 in Tonny A. Springer, Reductive groups. Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 3–27. We write $X^W$ for the subgroup of $W$-fixed points of $X$; then \begin{multline*} X^W=\{x\in X\ |\ s_\alpha(x)=x\ \ \forall \alpha\in S\}\\ =\{x\in X\ |\ \langle x,\alpha^\vee\rangle =0 \ \ \forall \alpha^\vee\in S^\vee\}= (S^\vee)^\bot. \end{multline*} Thus $X^*(G)\cong X_*(T^{\rm ss})^\bot = (S^\vee)^\bot=X^W$, as required.