Wonderful subgroups of reductive groups and spherical systems

TL;DR

This paper explores the relationship between wonderful subgroups of semisimple complex algebraic groups and their associated spherical systems, providing new insights and reducing the Luna conjecture to primitive cases.

Contribution

It establishes key results linking wonderful subgroups to their spherical systems and proves the existence of subgroups from combinatorial data, advancing the classification of G-varieties.

Findings

Proved relations between wonderful subgroups and spherical systems.

Reduced Luna conjecture to primitive cases.

Established existence of subgroups from combinatorial axioms.

Abstract

Let G be a semisimple complex algebraic group, and H a wonderful subgroup of G. We prove several results relating the subgroup H to the properties of a combinatorial invariant S of G/H, called its spherical system. It is also possible to consider a spherical system S as a datum defined by purely combinatorial axioms, and under certain circumstances our results prove the existence of a wonderful subgroup H associated to S. As a byproduct, we reduce for any group G the proof of the classification of wonderful G-varieties, known as the Luna conjecture, to its verification on a small family of cases, called primitive.