Weyl group of a compact Lie group vsWeyl group of a root system

Question

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $T$ be a maximal torus with Lie algebra $\mathfrak{t}$. I read that the Weyl group $W$ of $G$ is "the group of automorphisms of $T$ which are restrictions of inner automorphism of $G$": $N(T)/T$ where $N(T)$ is the normaliser of $T$. Now I have trouble seeing why/how this is related to the symmetries of the roots of the adjoint action of $\mathfrak{t}$ on the complexidied of $\mathfrak{g}$. Can someone help me here? Or is it a too general question? Thank you

Answer 1

The nLab article is correct but, of course, is very much incomplete (and it is not meant to be complete). The definition you should get out of their article is that $$ W≃N_G(T)/T, $$ as an abstract group. In fact, there is more to this formula, as it also gives you the standard linear representation of $W$: You let $N_G(T)$ act on the Lie algebra ${\mathfrak t}$ of $T$ via the adjoint representation (the linearization of the action of $N_G(T)$ on $T$ by conjugation). The kernel of this linear representation of $N_G(T)$ is exactly $T$ and, hence, you obtain the standard linear representation of $W$.

Connecting this to root systems is a very long story and many books are written on it. It is too long to be reproduced here. In brief: There is a canonical $W$-invariant (positive-definite) inner product on ${\mathfrak t}$, obtained by restricting the Killing form on ${\mathfrak g}$. One then studies codimension 1 subspaces (called "walls") in ${\mathfrak t}$ which are fixed by nontrivial elements of $W$. (In terms of the normalizer $N_G(T)$, the walls correspond to the codimension 1 subgroups of $T$ which have larger centralizer than the entire $T$.) These elements are involutions (since they are orthogonal transformations with codimension 1 fixed-point sets). The walls divide ${\mathfrak t}$ into connected components, called "chambers" and one proves that $W$ acts simply-transitively on the set of chambers. Then one picks one of the chambers $\Delta$ (called a positive chamber) and proves that $W$ is generated by reflections fixing the walls passing the codimension 1 faces of $\Delta$. These are called simple reflections in $W$. Roots of the root system of $G$ (or or ${\mathfrak g}$) are certain (carefully chosen) linear functionals on ${\mathfrak t}$ whose kernels are the walls in ${\mathfrak t}$. Then one verifies that these linear functionals satisfy a certain set of axioms, which then one uses to define an abstract root system. Later on, one relates this to the root system of the complex Lie algebra ${\mathfrak g}^{\mathbb C}$, the complexification of ${\mathfrak g}$. One can easily spend the entire semester getting to this point.

As for references, you can use the ones from the nLab or from this Wikipedia article.

This is a freely available set of notes on the subject which I found simply by googling.