Affine action

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

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

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

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