Using the notation of my previous question, let $N(T)$ denote the normalizer of the maximal torus $T$ and hence the Weyl group $W(G,T) = N(T)/T$.
Here think of roots $\alpha$ as maps $T \rightarrow GL(g_\alpha)$ where $g_\alpha$ is the corresponding root-space. Then I can see that the Weyl group has a permuting action on the roots buy conjugating their arguments but there seems to be some different ways of extending this action with non-trivial implications which I can't understand very well.
But is something stronger true about the above action? - as in for any root $\alpha$ there seems to exist a $s_\alpha \in W(G,T)$ such that it fixes $ker(\alpha)$ - what is such an action?
What is the action of $W(G,T)$ on $t$ ? (..which fixes the Stiefel diagram of the group consisting of the sets $exp^{-1}(ker(\alpha))$..are these hyperplanes in $t$?..why?..)
Let $R(T)$ be the representation ring of $T$. What is the action of $W(G,T)$ on $R(T)$? The point being that if I think of the obvious conjugation action of $W(G,T)$ on the arguments of $R(T)$ (..by conjugating the arguments..) then all elements in $R(T)$ seem to be $W(G,T)$-stable.
But it seems from literature that if one restricts the elements of the representation ring of $G$ to that of $T$ then the image lies in the $W(G,T)$ invariant subspace of $R(T)$ - which doesn't seem anything special since by the conjugation action everything in $R(T)$ is $W(G,T)$ stable! What am I missing?
- One defines the "weight lattice" as those weights in $t^*$ which evaluate to integers when evaluated on the co-roots of the simple roots. Now it seems that there is a natural action for the Weyl group on this weight lattice such that the lattice is invariant. What is this action?