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Journal of Lie Theory 23 (2013), No. 3, 607--638
Copyright Heldermann Verlag 2013

On the Local Structure Theorem and Equivariant Geometry of Cotangent Bundles

Vladimir S. Zhgoon
National Research University, Higher School of Economics, Vavilova 7, 117332 Moscow, Russia


Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We give a slightly improved version of Local Structure Theorems obtained by Knop and Timashev, which describe the action of some parabolic subgroup of $G$ on an open subset of $X$. We also extend various results of Vinberg and Timashev on the set of horospheres in $X$. We construct a family of nongeneric horospheres in $X$ and a variety ${\cal H}or_X$ parameterizing this family, such that there is a rational $G$-equivariant symplectic covering of cotangent vector bundles $T^*_{{\cal H}or_X}\rightarrow T^*_X$. As an application we recover the description of the image of the moment map of $T^*_X$ obtained by Knop. In our proofs we use only geometric methods which do not involve differential operators.

Keywords: Cotangent bundle, moment map, horosphere, Local Structure Theorem, little Weyl group.

MSC: 14L30; 53D05, 53D20

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