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.