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Let W {\displaystyle W} be the Weyl group of a semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} . Assume that a set of simple roots in h ∗ {\displaystyle {\mathfrak {h}}^{*}} is chosen.

The affine action of the Weyl group on the space h ∗ {\displaystyle {\mathfrak {h}}^{*}} is

where δ {\displaystyle \delta } is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.

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