Rationality patterns
Takuma Hayashi
TL;DR
The work develops a categorical framework for rationality problems in representation theory by extending Loewy’s real-form classification and Borel–Tits’ descent criterion to Γ-categories via descent data. It introduces Loewy data, Loewy pairs, and Loewy–Borel–Tits data to systematically study when self-conjugate simple objects admit $F$-forms and to identify minimal fields of definition, connecting these obstructions to endomorphism algebras and Brauer classes. The framework is then applied to $(\mathfrak{g},K)$-modules, proving that base-change preserves finite length, establishing local-global principles for BT cocycles, and giving methods to compute cocycles via $K$-types and to determine minimal fields for cohomological irreducible essentially unitarizable Harish-Chandra modules. These results unify descent theory, Galois cohomology, and representation theory, with potential geometric applications to equivariant twisted $D$-modules and broader rationality questions in algebraic representation theory.
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.