# Action of Weyl group on Lie algebra representation

Here is my set up.

• $\mathfrak{g}$: semisimple Lie algebra (assume base field has characteristic 0)
• $(\pi,V_\lambda)$: highest weight representation of $\mathfrak{g}$
• $W$: Weyl group of $\mathfrak{g}$, $w \in W$: arbitrary element
• $v_\lambda \in V_\lambda$ chosen highest weight vector.

Apparently it's true that $w \cdot v_\lambda$ has weight $w \cdot \lambda$.

How is $W$ acting on $v_\lambda$ (or $V_\lambda$) here? And why is this true?

• There is a natural projective representation of $W$ on $P(V_{\lambda})$ which satisfies this property (it comes from the restriction of $\pi$ to the normalizer of $T$). Not sure about a linear representation. Where did you see this statement? – Moishe Kohan Jul 1 '17 at 16:23