Affine action

Template:One source Let $W$ be the Weyl group of a semisimple Lie algebra $𝔤$ (associate to fixed choice of a Cartan subalgebra $𝔥$). Assume that a set of simple roots in ${𝔥}^{*}$ is chosen.

The affine action (also called the dot action) of the Weyl group on the space ${𝔥}^{*}$ is

$w\cdot \lambda :=w(\lambda +\delta )-\delta$

where $\delta$ is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.

References

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