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.
Categories: math.RT math.CT