The Chevalley Involution and a Duality of Weight Varieties

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dc.creatorHoward, Benjamin J.
dc.creatorMillson, John J.
dc.date2004-09-15
dc.date2004-09-22
dc.date.accessioned2026-07-07T09:49:11Z
dc.date.available2026-07-07T09:49:11Z
dc.descriptionWe 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.
dc.identifierhttps://arxiv.org/abs/math/0409260
dc.identifierhttp://arxiv.org/abs/math/0409260
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/164493
dc.subjectSymplectic Geometry
dc.subjectRepresentation Theory
dc.subject53D50
dc.titleThe Chevalley Involution and a Duality of Weight Varieties
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