# Affine action

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Let be the Weyl group of a semisimple Lie algebra (associate to fixed choice of a Cartan subalgebra ). Assume that a set of simple roots in is chosen.

The *affine action* (also called the *dot action*) of the Weyl group on the space is

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

## References

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