Journal of Lie Theory

no. 3
On the Local Structure Theorem and Equivariant Geometry of Cotangent Bundles
Journal of Lie Theory, Volume 23 (2013) no. 3, pp. 607-638
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.
DOI: 10.5802/jolt.740
Classification: 14L30, 53D05, 53D20
Keywords: Cotangent bundle, moment map, horosphere, Local Structure Theorem, little Weyl group 
@article{JOLT_2013_23_3_a0,
     author = {V. S. Zhgoon},
     title = {On the {Local} {Structure} {Theorem} and {Equivariant} {Geometry} of {Cotangent} {Bundles}},
     journal = {Journal of Lie Theory},
     pages = {607--638},
     year = {2013},
     volume = {23},
     number = {3},
     doi = {10.5802/jolt.740},
     zbl = {1284.14064},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.740/}
}
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V. S. Zhgoon. On the Local Structure Theorem and Equivariant Geometry of Cotangent Bundles. Journal of Lie Theory, Volume 23 (2013) no. 3, pp. 607-638. doi: 10.5802/jolt.740

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