Rational structures on quivers and a generalization of Gelfand's equivalence

Abstract

We introduce the notion of rational structure on a quiver and associated representations to establish a coherent framework for studying quiver representations in separable field extensions. This notion is linked to a refinement of the notion of $K$-species, which we term \'etale $K$-species: We establish a categorical anti-equivalence between the category of $K$-rational quivers and that of \'etale $K$-species, which extends to an equivalence of their respective representation categories. For $K$-rational quivers there is a canonical notion of base change, which suggests a corresponding notion of base change for (\'etale) $K$-species which we elaborate. As a primary application, we generalize Gelfand's celebrated equivalence between certain blocks of Harish-Chandra modules for $\mathrm{SL}_2(\mathbb{R})$ and representations of the Gelfand quiver to a rational setting. To this end, we define a $\mathbb{Q}$-rational structure on the Gelfand quiver and its representations. A key technical tool, which we call unipotent stabilization, is developed to construct the functor from certain rational Harish-Chandra modules to nilpotent rational quiver representations. We prove that this functor is an equivalence. A similar result is established for the cyclic quiver. A notable consequence of this rational framework is that the defining relation of the Gelfand quiver becomes superfluous when working over fields not containing $\sqrt{-1}$. This allows us to recast our results in the language of $\mathbb{Q}$-species without relations.