Normal structure of isotropic reductive groups over rings

TL;DR

This paper establishes a classification of subgroups normalized by elementary subgroups in isotropic reductive groups over rings, revealing a structured lattice parametrized by ideals, under specific algebraic conditions.

Contribution

It proves a sandwich classification theorem for subgroup lattices in isotropic reductive groups over rings, extending understanding of their subgroup structure in algebraic group theory.

Findings

Lattice of subgroups splits into disjoint sandwiches parametrized by ideals

Elementary subgroup E(R,q) and centralizer C(R,q) form the sandwich structure

The theorem applies to groups over arbitrary rings with invertible structure constants

Abstract

The paper studies the lattice of subgroups of an isotropic reductive group G(R) over a commutative ring R, normalized by the elementary subgroup E(R). We prove the sandwich classification theorem for this lattice under the assumptions that the reductive group scheme G is defined over an arbitrary commutative ring, its isotropic rank is at least 2, and the structure constants are invertible in R. The theorem asserts that the lattice splits into a disjoint union of sublattices (sandwiches) E(R,q)<=...<=C(R,q) parametrized by the ideals q of R, where E(R,q) denotes the relative elementary subgroup and C(R,q) is the inverse image of the center under the natural homomorphism G(R) to G(R/I). The main ingredients of the proof are the "level computation" by the first author and the universal localization method developed by the second author.