arXiv2026
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 étale $K$-species. We establish a categorical anti-equivalence between $K$-rational quivers and étale $K$-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 (étale) $K$-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 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. Both results hold over every field $E$ of characteristic $0$, for suitable $E$-rational structures on the two quivers; for $E=\mathbb{C}$ they recover Gelfand's equivalence. In this rational framework the defining relation of the Gelfand quiver acquires a descent-theoretic meaning: its Galois orbit is stable, it descends to a single relation on the associated $\mathbb{Q}$-species, and a $\mathbb{Q}$-rational representation satisfies it if and only if the composite of the two arrows through the vertex $-$ is a $\mathbb{Q}$-rational endomorphism of the space at the central vertex. This allows us to recast our results in the language of $\mathbb{Q}$-species.