  We study the action of the Chevalley involution of a simple complex Lie group
G on the set of its weight varieties (i.e. torus quotients of its flag
manifolds). We find for torus quotients of Grassmannians (for GL(n)) that we
obtain the classical notion of "association of projective point sets" (see Ch.
III of Dolgachev-Ortland, Asterisque 165). We say a weight variety is
"self-dual" if it is carried into itself by the action of the Chevalley
involution. Our main theorem is a classification of those self-dual weight
varieties for which the induced self-map is the identity. This problem is the
"classical problem" associated to the "quantum problem" of classifying the
self-dual representations V of G for which the induced action of the Chevalley
involution on the zero weight space V[0] is a scalar. This latter problem is
important in the study of the representation of the Artin group associated to G
on V[0] obtained as the monodromy representation of the Casimir connection -
see Millson - Toledano Laredo arXiv:math.QA/0305062.
