Chevalley对合与权簇的对偶
The Chevalley Involution and a Duality of Weight Varieties
摘要：
研究了单复李群G的Chevalley对合在其权簇（即旗流形的环面对合）集合上的作用.我们发现，对于格拉斯曼环（GL（n）），我们得到了经典的概念“协会的射影点集”（见第三章Dolgachev-Ortland，Asterisque 165）。我们说一个权簇是“自对偶的”，如果它是由Chevalley对合的作用带入自身的。我们的主要定理是一个分类的自对偶的重量品种，其诱导自映射是单位。这个问题是与对G的自对偶表示V进行分类的“量子问题”相关联的“经典问题”，其中Chevalley对合在零权空间V[0]上的诱导作用是标量。后一个问题在研究与V[0]上的G相关的Artin群的表示时很重要，该表示作为Casimir连接的单值表示获得-参见Millson - Toledano Laredo arXiv：math.QA/0305062。
英文摘要：
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.
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