Let W {\displaystyle W} be the Weyl group of a semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} (associate to fixed choice of a Cartan subalgebra h {\displaystyle {\mathfrak {h}}} ). Assume that a set of simple roots in h ∗ {\displaystyle {\mathfrak {h}}^{*}} is chosen. The affine action (also called the dot action) of the Weyl group on the space h ∗ {\displaystyle {\mathfrak {h}}^{*}} is
w ⋅ λ := w ( λ + δ ) − δ {\displaystyle w\cdot \lambda :=w(\lambda +\delta )-\delta }
where δ {\displaystyle \delta } is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.
References Baston, Robert J.; Eastwood, Michael G. (1989), The Penrose Transform: its Interaction with Representation Theory, Oxford University Press.