Invariant forms on irreducible modules of simple algebraic groups

TL;DR

This paper investigates the existence of non-degenerate invariant quadratic and alternating forms on irreducible modules of simple algebraic groups, providing solutions in classical and some exceptional cases, and refining subgroup classifications.

Contribution

It solves the classical problem of determining invariant quadratic forms for self-dual modules in characteristic 2, and refines the classification of maximal subgroups of simple algebraic groups.

Findings

Characterizes when self-dual modules have invariant quadratic forms in characteristic 2.

Provides explicit criteria for fundamental weights and certain sums of weights.

Refines Seitz's classification of maximal subgroups in classical groups.

Abstract

Let $G$ be a simple linear algebraic group over an algebraically closed field $K$ of characteristic $p \geq 0$ and let $V$ be an irreducible rational $G$-module with highest weight $\lambda$. When $V$ is self-dual, a basic question to ask is whether $V$ has a non-degenerate $G$-invariant alternating bilinear form or a non-degenerate $G$-invariant quadratic form. If $p \neq 2$, the answer is well known and easily described in terms of $\lambda$. In the case where $p = 2$, we know that if $V$ is self-dual, it always has a non-degenerate $G$-invariant alternating bilinear form. However, determining when $V$ has a non-degenerate $G$-invariant quadratic form is a classical problem that still remains open. We solve the problem in the case where $G$ is of classical type and $\lambda$ is a fundamental highest weight $\omega_i$, and in the case where $G$ is of type $A_l$ and $\lambda = \omega_r + \omega_s$ for $1 \leq r < s \leq l$. We also give a solution in some specific cases when $G$ is of exceptional type. As an application of our results, we refine Seitz's $1987$ description of maximal subgroups of simple algebraic groups of classical type. One consequence of this is the following result. If $X < Y < \operatorname{SL}(V)$ are simple algebraic groups and $V \downarrow X$ is irreducible, then one of the following holds: (1) $V \downarrow Y$ is not self-dual; (2) both or neither of the modules $V \downarrow Y$ and $V \downarrow X$ have a non-degenerate invariant quadratic form; (3) $p = 2$, $X = \operatorname{SO}(V)$, and $Y = \operatorname{Sp}(V)$.