Representation theory of disconnected reductive groups

TL;DR

This paper extends key aspects of representation theory from connected reductive groups to disconnected reductive groups, covering classification, module properties, and characteristic-related decomposition maps.

Contribution

It provides the first comprehensive generalization of fundamental representation theory results to disconnected reductive groups.

Findings

Classification of irreducible representations established

Existence and properties of Weyl modules confirmed

Decomposition maps between characteristics analyzed

Abstract

We study three fundamental topics in the representation theory of disconnected algebraic groups whose identity component is reductive: (i) the classification of irreducible representations; (ii) the existence and properties of Weyl and dual Weyl modules; and (iii) the decomposition map relating representations in characteristic $0$ and those in characteristic $p$ (for groups defined over discrete valuation rings). For each of these topics, we obtain natural generalizations of the well-known results for connected reductive groups.