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# Affine action

V. de Paiva. "A dialectica-like model of linear logic". In Proc. Conf. on Category Theory and Computer Science, Springer-Verlag Lecture Notes in Computer Science 389, pp. 341–356, Manchester, September 1989.

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.

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