Exceptional representations of simple algebraic groups in prime characteristic

TL;DR

This paper classifies irreducible exceptional modules over simple algebraic groups in prime characteristic, identifying conditions and providing a complete classification for exceptional types, with reductions for classical types.

Contribution

It introduces a necessary condition for modules to be exceptional and completes the classification for exceptional types, reducing the problem for classical types to a short list of unclassified modules.

Findings

Complete classification for exceptional type modules.

Reduction of classical type modules to a short list.

Identification of a necessary condition for exceptionality.

Abstract

Let G be a simply connected simple algebraic group over an algebraically closed field K of characteristic p>0 with root system R, and let ${\mathfrak g}={\cal L}(G)$ be its restricted Lie algebra. Let V be a finite dimensional ${\mathfrak g}$-module over K. For any point $v\inV$, the {\it isotropy subalgebra} of $v$ in $\mathfrak g$ is ${\mathfrak g}_v=\{x\in{\mathfrak g}/x\cdot v=0\}$. A restricted ${\mathfrak g}$-module V is called exceptional if for each $v\in V$ the isotropy subalgebra ${\mathfrak g}_v$ contains a non-central element (that is, ${\mathfrak g}_v\not\subseteq {\mathfrak z(\mathfrak g)}$). This work is devoted to classifying irreducible exceptional $\mathfrak g$-modules. A necessary condition for a $\mathfrak g$-module to be exceptional is found and a complete classification of modules over groups of exceptional type is obtained. For modules over groups of classical type, the general problem is reduced to a short list of unclassified modules. The classification of exceptional modules is expected to have applications in modular invariant theory and in classifying modular simple Lie superalgebras.