This is a classical result in the theory of Lie groups, but I am stuck on trying to understand the action of the Weyl group on the classifying space of the torus. Namely,
Let $G$ be a compact connected Lie Group and $T$ be a maximal torus in $G$. Let $W = N(T)/T$ be the Weyl group of $T$ in $G$. This groups acts on $T$ by conjugation (the action is well defined as $T$ is abelian) and so it induces an action on the classifying space $BT$.
This is the argument that I still do not completely get
Mimura M, Toda H. Topology of Lie groups, I and II. American Mathematical Soc.; 1991.
Let $p: BT \rightarrow BG$ be the map induced by the inclusion, and $f_n : BT \rightarrow BT$ the map induced by the conjugation of $nT \in W$. Then $f_n(eT) = enT$ for $e \in ET$ and so $$(p\circ f_n)(eT) = p(enT) = p(enG) = eG = p(eT)$$ Therefore, $p^*(H^*(BG)) \subseteq H^*(BT)^W$
It is not clear for me why the conjugation action induces the map $f_n$ defined above. I'd appreciate if someone has an insight of how to compute the map out of the action.