How does the weyl group act on weights\roots

Question

Let the Weyl group be: $$W=N(T)/T$$ where $T$ is the maximal torus of some lie group $G$ and $N(T)$ is the normalizer of $T$. I saw that in this question that the Weyl group acts on weights by: $$(w.\chi)(t)=\chi(wtw^{-1})$$ but as far as I know weights (are characters and thus) are class functions. This will mean $w.\chi = \chi$.

Am I missing something? I looked in other sources but none write the action of the Weyl group explicitly.

Yes: the $\chi$ in that question are characters of $T$, not of $W$. — Matthew Towers, Jun 09 '22 at 10:30
@MatthewTowers Oh thanks :) I felt like I was missing something simple. Could you post it as an answer so I'll accept it? — Alexey, Jun 09 '22 at 10:31

score 1 · Accepted Answer · answered Jun 09 '22 at 10:33

1

The characters $\chi$ are characters of $T$, not of $W$, so they're not class functions for $W$ - in fact they're not even defined on elements of $W$.

answered Jun 09 '22 at 10:33

Matthew Towers

10,854

How does the weyl group act on weights\roots

1 Answers1