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.
References
- Baston, Robert J. & Eastwood, Michael G. (1989), "The Penrose Transform: its Interaction with Representation Theory", Oxford University Press.