On the Local Structure Theorem and Equivariant Geometry of Cotangent Bundles

Vladimir S. Zhgoon¹

¹National Research University, Higher School of Economics, Vavilova 7, 117332 Moscow, Russia

Journal of Lie Theory, Tome 23 (2013) no. 3, pp. 607-638

Résumé

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.

Détail

Classification : 14L30, 53D05, 53D20
Mots-clés : Cotangent bundle, moment map, horosphere, Local Structure Theorem, little Weyl group