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Selecta Mathematica

, Volume 21, Issue 3, pp 931–993 | Cite as

Strongly solvable spherical subgroups and their combinatorial invariants

  • Roman AvdeevEmail author
Article

Abstract

A subgroup \(H\) of an algebraic group \(G\) is said to be strongly solvable if \(H\) is contained in a Borel subgroup of \(G\). This paper is devoted to establishing relationships between the following three combinatorial classifications of strongly solvable spherical subgroups in reductive complex algebraic groups: Luna’s general classification of arbitrary spherical subgroups restricted to the strongly solvable case, Luna’s 1993 classification of strongly solvable wonderful subgroups, and the author’s 2011 classification of strongly solvable spherical subgroups. We give a detailed presentation of all the three classifications and exhibit interrelations between the corresponding combinatorial invariants, which enables one to pass from one of these classifications to any other.

Keywords

Algebraic group Homogeneous space Representation  Spherical variety Wonderful variety Spherical subgroup Solvable subgroup 

Mathematics Subject Classification

14M27 14M17 05E15 

Notes

Acknowledgments

The author cordially thanks D. Luna for discussions and private notes, which among other things helped the author to learn much about Luna’s general classification of spherical homogeneous spaces. Thanks are also due to I. V. Arzhantsev, P. Bravi, S. Cupit-Foutou, D. A. Timashev, E. B. Vinberg, and V. S. Zhgoon for helpful discussions on particular topics. At last, the author is grateful to the referee for numerous corrections and valuable suggestions.

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Authors and Affiliations

  1. 1.National Research University Higher School of EconomicsMoscowRussia

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