We have the following setup: $G$ a compact reductive group over $\mathbb{R}$, $T$ a maximal Torus, $U^1$ the real algebraic Torus and $W$ the Weil group of $T$
In Variétiés de Shimura Lemma 1.2.4 p.258 Deligne concludes the following isomorphism:
$$ Hom(U^1,G)/G(\mathbb{R})\cong Hom(\mathbb{G}_m, G_\mathbb{C})/G_\mathbb{C}(\mathbb{C})$$ from these isomorphisms: $$ Hom(U^1,T)/W\cong Hom(U^1,G)/G(R)\\ Hom(\mathbb{G}_{m},T_\mathbb{C})/W\cong Hom(\mathbb{G}_m,G_\mathbb{C})/G_\mathbb{C}(\mathbb{C})$$
- What is the "Weil group of T"? I am aware of the definition of a Weil group of a local field, maybe he just means the Weil group of the local field over which $T$ is defined?
- I do not know how to conclude.
I know any two maximal tori are conjugate and that the image of a torus will be contained in a torus, which will give me e.g. a map $Hom(\mathbb{G}_{m},T_\mathbb{C})\to Hom(\mathbb{G}_m,G_\mathbb{C})/G_\mathbb{C}(\mathbb{C})$, which is surjective, but from there I am stuck. Is he maybe referring to the Weyl group of $T$?
Given the two latter isomorphisms, how do I conclude the first one? Is there an "obvious" isomorphism $Hom(U^1,T)/W\to Hom(\mathbb{G}_{m},T_\mathbb{C})/W$ given by complexification?
I would also welcome a reference which possibly elaborates on such isomorphisms between tori and reductive groups.