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Chebyshevskii Sb., 2015, Volume 16, Issue 4, Pages 164–187 (Mi cheb440)  

Little Weyl groups and variety of degenerate horospheres

V. S. Zhgoonab

a Scientific Research Institute for System Studies of RAS, Moscow
b National Research University "Higher School of Economics" (HSE), Moscow

Abstract: Let $G$ be a connected reductive group acting on an irreducible normal algebraic variety $X$. We study equivariant geometry of the cotangent vector bundle $X$, and we apply these results to study of a little Weyl group. The aim of this paper is to extend various results of E. B. Vinberg, who constructed a rational Galois cover of $T^*_X$ of quasiaffine $X$ by means of cotangent bundle to the so-called variety of generic horosheres. It is well-known that the example of a flag variety shows that these results could not be generalized directly.
We develop the results of D. A. Timashev [18], who obtained the generalizations of the results of Vinberg to the class of varieties wider than quasiaffine but smaller than quasiprojective.
We construct a family of horospheres of a smaller dimension in $X$ which are called degenerate, and a variety $\mathcal{H}or$ parameterizing this family, which has the same dimension as the variety parametrizing generic horosheres. Moreover in the quasiaffine case our construction shows that the familly of degenerate horosheres coincides with the familly of generic ones.
We show that for constructed family of horosheres there exists a rational $G$-equivariant symplectic Galois covering of cotangent vector bundles $T^*_{\mathcal{H}or} \dashrightarrow T^*_X$. It is proved that the extension of the fields of rational functions corresponding to this cover is a finite Galois extension with the Galois group isomorphic to the little Weyl group.
As an application we get the description of the image of the moment map of $T^*_X$ and the image of the normalized moment map by means of purely geometric methods. The first description of the image of the normalized moment map was obtained by F. Knop, nevertheless his proof is non-elementary since it involves the methods of differential operators.
Bibliography: 21 titles.

Keywords: cotangent bundle, moment map, horosphere, local structure theorem, little weyl group.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-02094 а
13-01-12402 офи-м2
Russian Science Foundation 14-21-00052
Ministry of Education and Science of the Russian Federation 5/100

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UDC: 512.745.2
Received: 06.03.2015

Citation: V. S. Zhgoon, “Little Weyl groups and variety of degenerate horospheres”, Chebyshevskii Sb., 16:4 (2015), 164–187

Citation in format AMSBIB
\by V.~S.~Zhgoon
\paper Little Weyl groups and variety of degenerate horospheres
\jour Chebyshevskii Sb.
\yr 2015
\vol 16
\issue 4
\pages 164--187

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