Affine action

Let formula_1 be the Weyl group of a semisimple Lie algebra formula_2 (associate to fixed choice of a Cartan subalgebra formula_3). Assume that a set of simple roots in formula_4 is chosen. 
The "affine action" (also called the "dot action") of the Weyl group on the space formula_4 is
where formula_7 is the sum of all fundamental weights, or, equivalently, the half of the sum of all positive roots.