Easy proof that reflections generate $N(T)/T$ for connected compact group?

Question

I'm teaching a course on Coxeter groups and I'd like to provide an overview of the connection to compact Lie groups. Let $G$ be a compact connected Lie group, $T$ a maximal torus and $N(T)$ the normalizer. Then we can define the Weyl group $W$ as $N(T)/T$. Since $G$ is compact, the conjugation action preserves an inner product on $\mathfrak{g}$, so $W$ preserves an inner product on $\mathfrak{t}$.

Each root pair $(\pm \alpha)$ gives a map $\phi: SU(2) \to G$, and $\phi\left( \begin{smallmatrix} 0&-1 \\ 1&0 \end{smallmatrix} \right)$ lands in $N(T)$ and gives the reflection in $\alpha$. So we have a lot of orthogonal reflections inside $W$.

Is there an easy way to see that these orthogonal reflections generate $W$ defined as $N(T)/T$? I'm glad to use structural properties of reflection groups, since this is the actual topic of the course, but I certainly don't have time to develop the full Lie structure theory.

Answer 1

Here are some further comments, in community-wiki format. As earlier comments indicate, compact Lie groups and their maximal tori raise many questions. For instance, the real case is perhaps more basic (and also more familiar to most students from their linear algebra acquaintance with orthogonal matrices, etc.). Complexification mainly connects the subject with Lie algebra theory. But you'd clearly prefer to avoid exceptional types and such.

One possible approach is to rely on the most familiar examples: O(n) and SO(n). In these cases, there is the old theorem of Dieudonne, asserting that every group element is a product of at most n "symmetries". But since a reflection in this concrete setting has determinant -1, there is an extra step in passing to N(T)/T in the connected Lie group SO(n).

Beyond matrix theory examples, a delicate issue in relating compact Lie groups and (finite) Coxeter groups is that not all of the latter finite groups are Weyl groups coming from conventional root systems, e.g., most dihedral groups. The classifications in Lie theory and Coxeter group theory are close but not identical. Maybe the optimal connection is seen in the context of work by Tits, usually involving BN-pair language. Coxeter himself was a geometer who studied especially finite and affine groups generated by reflections, but Bourbaki and Tits developed more of the Lie theoretic connection after early work of Witt while broadening the idea of "Coxeter group".

Good luck!

Answer 2

$\def\Wr_{W_{refl}}$Here is an approach I could gesture towards, though I certainly couldn't complete. Let $\Wr$ be the subgroup generated by reflections, so we have $\Wr \to N(T)/Z(T) \leftarrow N(T)/T$. Let's concentrate on showing the left arrow is an isomorphism; I have no insight into the right arrow.

In other words, we need to show that two elements of $T$ are $G$-conjugate if and only if they are $\Wr$-conjugate. It suffices to show that every $\Wr$-invariant function on $T$ is a class function. Letting $\Lambda$ be the character lattice of $T$, the invariant ring $\mathbb{C}[\Lambda]^{\Wr}$ has a basis indexed by dominant characters. On the other hand, traces of representations of $G$ are clearly class functions. So this result follows from the fact that every dominant character is the highest weight of a $G$-representation.

I like this because I was already planning to do invariant theory of Coxeter groups so $\mathbb{C}[\Lambda]^{\Wr}$ ties back to the course, and because I do know a construction of the irreps of $G$ which directly takes a dominant weight and build a representation. That construction is very complex analytic -- Borel subgroups, Bruhat decomposition and Hartog's lemma -- but at least I won't be lying when I say it can be done. And maybe one of you knows a better way!

Answer 3

Here are some remarks that may be useful. Consider the homogeneous space $X=G/T$ with the torus acting on the left $T\circlearrowright X$. Then it's easy to see that the $T$ fixed points are exactly the cosets in $W=N(T)/T$. At each fixed point $w\in W$, the tangent space $T_wX$ defines a (complex) representation of $T$, which splits into a direct sum of $2$-(real) dimensional weight spaces $T_wX=\bigoplus_{\alpha\in\Phi^+} L_{w^{-1}(\alpha)}$. Define the $2$-dimensional compact submanifolds $P_{w^{-1}(\alpha)}=\operatorname{exp}(L_{w^{-1}(\alpha)})\subset X$. Perhaps it's also not difficult to see (or at least believe) that $P_{w^{-1}(\alpha)}$ is a $T$-invariant $\mathbb{S}^2$, containing exactly two fixed points. Now you have a graph $\Gamma=\left(W,\left\{P_{w^{-1}(\alpha)}\right\}\right)$.

$\textbf{Claim:}$ The graph $\Gamma$ is connected.

I don't really have an easy proof of this. One idea would be to use the convexity of the moment mapping $\mu\colon X\rightarrow\mathfrak{t}^*$, i.e. the image of the moment map is a convex polytope whose graph is a subgraph of $\Gamma$ (containing all vertices). Then recall that the graph of a convex polytope is always connected...

For each index $w^{-1}(\alpha)$, $\alpha\in \Phi^+$, define $s_{w^{-1}(\alpha)}\in W$ as the element $w^{-1}\cdot w'$ where $P_{w^{-1}(\alpha)}$ is the edge joining $w$ and $w'$. Then from the connectivity of $\Gamma$, it's easy to see that $W$ must be generated by $\mathcal{T}=\left\{s_{w^{-1}(\alpha)}\left|w\in W, \ \alpha\in\Phi^+\right.\right\}$. Of course it only remains to show that each of these elements in $\mathcal{T}$ is a reflection. To see this just remark that $s_{w^{-1}(\alpha)}$ must preserve the (kernel of the) weight $\lambda_{w^{-1}(\alpha)}\colon\mathfrak{t}\rightarrow \mathbb{R}$ defining the weight space $L_{w^{-1}(\alpha)}$. And of course since it fixes a hyperplane, it must be a reflection.