Representations of general linear groups and categorical actions of Kac-Moody algebras

TL;DR

This paper explores categorical actions of Kac-Moody algebras on categories of rational representations of general linear groups, highlighting new structures and their connections to crystals and polynomial representations.

Contribution

It introduces the concept of categorical Kac-Moody actions through the example of general linear groups in positive characteristic, extending previous theories.

Findings

Categorical actions are constructed on rational representations of GL in positive characteristic.

Connections between categorical actions and crystal structures are established.

Advanced topics include categorical actions on polynomial representations.

Abstract

This is an expanded version of the lectures given by the author on the 3rd school "Lie algebras, algebraic groups and invariant theory" in Togliatti, Russia. In these notes we explain the concept of a categorical Kac-Moody action by studying an example of the category of rational representations of a general linear group in positive characteristic. We also deal with some more advanced topics: a categorical action on the polynomial representations and crystals of categorical actions.