Rationality patterns

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.

Final version. Section numbers were changed. The terminology "strongly'' was replaced with "strong'', for example, Definition 4.4.11 (3). Other minor typos were corrected. To appear in Journal of Algebra

References in corpus (4)

• Galois and Cartan Cohomology of Real Groups
• Flat Base Change Formulas for $(\mathfrak{g},K)$-modules over Noetherian rings
• Half-integrality of line bundles on partial flag schemes of classical Lie groups
• Classification of irreducible representations of affine group superschemes and the division superalgebras of their endomorphisms