Affine action

Let [math]\displaystyle{ W }[/math] be the Weyl group of a semisimple Lie algebra [math]\displaystyle{ \mathfrak{g} }[/math] (associate to fixed choice of a Cartan subalgebra [math]\displaystyle{ \mathfrak{h} }[/math]). Assume that a set of simple roots in [math]\displaystyle{ \mathfrak{h}^* }[/math] is chosen. The affine action (also called the dot action) of the Weyl group on the space [math]\displaystyle{ \mathfrak{h}^* }[/math] is

[math]\displaystyle{ w\cdot \lambda:=w(\lambda+\delta)-\delta }[/math]

where [math]\displaystyle{ \delta }[/math] 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 .