Compact Weyl group

Let $G$ be a compact connected Lie group and let $T\le G$ be a maximal torus (Tori and maximal tori). The normalizer of $T$ in $G$ is $N_G(T)=\{g\in G:gT{g}^{-1}=T\},$ a subgroup of $G$ containing $T$; since $T$ is closed and conjugation is continuous, $N_G(T)$ is closed in $G$, hence compact. The Weyl group of the pair $(G,T)$ is the quotient group $W(G,T):=N_G(T)\mathrm{/}T\mathrm{.}$ It is a group because $T$ is a normal subgroup of $N_G(T)$, and it acts on $T$ by $(gT)\cdot t:=gt{g}^{-1},$ a well-defined action: replacing $g$ by $g{t}^{\mathrm{\prime }}$ with $t^{\mathrm{\prime }}\in T$ changes $gt{g}^{-1}$ to $(g{t}^{\mathrm{\prime }})t(g{t}^{\mathrm{\prime }}{)}^{-1}=g(t^{\mathrm{\prime }}t{t}^{\mathrm{\prime }-1})g^{-1}=gt{g}^{-1}$ because $T$ is abelian. The differential of this action at the identity is the linear action of $gT$ on $\mathfrak{t}=\mathrm{Lie}(T)$ by ${\mathrm{Ad}}_g{\mathrm{\mid }}_{\mathfrak{t}}$, and the action on $\mathfrak{t}$ is a homomorphism from $W(G,T)$ to $\mathrm{GL}(\mathfrak{t})$.

The Weyl group is defined relative to the chosen maximal torus; a conjugacy $gT{g}^{-1}=T^{\mathrm{\prime }}$ identifies $W(G,T)$ with $W(G,T^{\mathrm{\prime }})$ by $w\mapsto gw{g}^{-1}$, so the isomorphism type of $W(G,T)$ does not depend on the choice of maximal torus up to conjugacy.

• The Weyl group is a group of automorphisms of the torus in this definition; it is not yet asserted to be finite, nor identified with the reflection group of a root system. Those are theorems proved on this page.
• An element $w=gT\in W(G,T)$ is trivial exactly when $g\in T$; the kernel of the action on $T$ is computed on this page when $W(G,T)$ is proved to be equal to the centralizer quotient.