# Affine action

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.