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?