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Modality of representations

arxiv.org. math. Cornell University, 2017. No. arXiv:1707.07720v1 [math.RT] 24 Jul 2017.
We first establish several general properties of modality of al- gebraic group actions. In particular, we introduce the notion of a modali- ty-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 < 3. Next, ex- ploring 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.