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Abstract

Abstract. We develop a structure theory of connected solvable spherical subgroups in semisimple algebraic groups. Based on this theory, we obtain an explicit classification of all such subgroups up to conjugacy. §1. Introduction 1.1. Let G be a connected semisimple complex algebraic group. A closed subgroup H ⊂ G (resp. a homogeneous space G/H) is said to be spherical if one of the following three equivalent conditions holds:( 1) There are other characterizations of spherical subgroups, but the three given above are most often used to study them.Spherical homogeneous spaces have been studied intensively over the last thirty years. However, the problem of classifying these spaces or, equivalently, classifying spherical subgroups in semisimple algebraic groups still remains of importance. Let us give a brief historical account of this question. The first significant result in this direction was obtained by Krämer [1] in 1979. He classified all reductive spherical subgroups in simple groups. Then Mikityuk [2] in 1986 and, independently, Brion [3] in 1987 classified all reductive spherical subgroups in arbitrary semisimple groups (see also [4] for a more accurate formulation). The next step towards a classification of spherical homogeneous spaces was Luna's 1993 preprint [5], where solvable spherical subgroups in semisimple groups were considered. In this preprint, under certain restrictions, all such subgroups were described in the following sense: with each subgroup one associates a set of combinatorial data that uniquely determines this subgroup, and then one classifies all sets that can appear in this way. In 2001 Luna [6] created a theory of spherical systems and, using this theory, described (in the same sense) all spherical subgroups in semisimple groups of type A. During the following several years Luna's approach was applied successfully by Bravi and Pezzini to certain other types of semisimple groups, including all the classical groups (for details, the reader is referred to the paper [7] and its bibliography). Finally, in 2009 a new approach to the problem was proposed by CupitFoutou who completed the proof of the so-called Luna conjecture and thereby obtained a description of all spherical subgroups in arbitrary semisimple groups [8]. Thus, at