Affine action

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

The affine action (also called the dot action) of the Weyl group on the space \mathfrak{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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