On irreducible subgroups of simple algebraic groups

TL;DR

This paper investigates the structure of irreducible subgroups within simple algebraic groups, focusing on classical groups and reducing classification to specific cases involving orthogonal groups and spin modules.

Contribution

It extends previous classification results by analyzing triples (G, H, V) where G is classical, H is positive-dimensional, and V is a specific irreducible module, narrowing down the classification to orthogonal groups and spin modules.

Findings

Reduces classification of triples to orthogonal groups with spin modules.

Identifies conditions under which subgroups preserve orthogonal decompositions.

Builds on prior work to refine understanding of subgroup structures in algebraic groups.

Abstract

Let $G$ be a simple algebraic group over an algebraically closed field $K$ of characteristic $p\geqslant 0$, let $H$ be a proper closed subgroup of $G$ and let $V$ be a nontrivial irreducible $KG$-module, which is $p$-restricted, tensor indecomposable and rational. Assume that the restriction of $V$ to $H$ is irreducible. In this paper, we study the triples $(G,H,V)$ of this form when $G$ is a classical group and $H$ is positive-dimensional. Combined with earlier work of Dynkin, Seitz, Testerman and others, our main theorem reduces the problem of classifying the triples $(G,H,V)$ to the case where $G$ is an orthogonal group, $V$ is a spin module and $H$ normalizes an orthogonal decomposition of the natural $KG$-module.