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Article

On the Local Structure Theorem and equivariant geometry of cotangent bundles

Journal of Lie Theory. 2013. Vol. 23. P. 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 $\Hor$ parameterizing this family, such that there is  a rational  $G$-equivariant symplectic covering  of cotangent vector bundles $T^*\Hor \dashrightarrow T^*X$. 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.