Rationality patterns

TL;DR

This paper develops categorical frameworks extending classical classification schemes for representations of algebraic groups, with applications to Harish-Chandra modules and their fields of definition.

Contribution

It introduces general categorical formalisms that extend Loewy's classification and Borel--Tits' criterion, applied to the theory of Harish-Chandra modules over various fields.

Findings

Extended classification schemes for representations

Criteria for rational forms of representations

Applications to fields of definition of Harish-Chandra modules

Abstract

In this paper, we establish general categorical frameworks that extend Loewy's classification scheme for finite-dimensional real irreducible representations of groups and Borel--Tits' criterion for the existence of rational forms of representations of $\bar{F}\otimes_F G$ for a connected reductive algebraic group $G$ over a field $F$ of characteristic zero and its algebraic closure $\bar{F}$. We also discuss applications of these general formalisms to the theory of Harish-Chandra modules, specifically to classify irreducible Harish-Chandra modules over fields $F$ of characteristic zero and to identify smaller fields of definition of irreducible Harish-Chandra modules over $\bar{F}$, particularly in the case of cohomological irreducible essentially unitarizable modules.