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?

  • 2
    $\begingroup$ 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? $\endgroup$ – Moishe Kohan Jul 1 '17 at 16:23

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