Modality of representations

TL;DR

This paper studies the properties of modality in algebraic group actions, introduces modality-regular actions, classifies certain representations, and generalizes the concept of packets in Lie algebra actions.

Contribution

It establishes general properties of modality, introduces modality-regular actions, and classifies irreducible representations of simple algebraic groups with low modality, also generalizing packets in Lie algebra actions.

Findings

Every visible action is modality-regular.

Classified irreducible linear representations with modality ≤ 2.

Generalized the notion of packets to cyclically graded semisimple Lie algebras.

Abstract

We first establish several general properties of modality of algebraic group actions. In particular, we introduce the notion of a modality-regular action and prove that every visible action is modality-regular. Then, using these results, we classify irreducible linear representations of connected simple algebraic groups of every fixed modality $\leqslant 2$. Next, exploring a finer geometric structure of linear actions, we generalize to the case of any cyclically graded semisimple Lie algebra the notion of a packet (or a Jordan/decomposition class) and establish the properties of packets.