affine action

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

{{citation|first1=Robert J.|last1=Baston|first2=Michael G.|last2=Eastwood|authorlink2=Michael Eastwood|title=The Penrose Transform: its Interaction with Representation Theory|publisher=Oxford University Press|year=1989}}.