Representation theory of the Gelfand quiver and Harish-Chandra modules for $\mathsf{SL}_2(\mathbb{R})$

TL;DR

This paper classifies indecomposable objects in the derived category of the Gelfand quiver related to Harish-Chandra modules for SL(2,R), using combinatorial and Lie-theoretic methods.

Contribution

It provides a complete classification of indecomposables in the derived category, extending previous solutions to a derived category perspective.

Findings

Classified indecomposable objects as band and string complexes.

Determined images under derived Auslander-Reiten translation and dualities.

Provided explicit projective resolutions and homological invariants.

Abstract

In 1970, Gelfand posed the problem of classifying the indecomposable objects in a representation category equivalent to the principal block of Harish-Chandra modules for $\mathsf{SL}_2(\mathbb{R})$; explicit solutions were obtained by Bondarenko, and, independently, Crawley-Boevey. In this article, we give a complete answer to Gelfand's problem from a derived category perspective. We classify indecomposable objects in the bounded derived category of nilpotent representations of the Gelfand quiver in terms of band and string complexes, and determine their images under the derived Auslander-Reiten translation, the sign involution, and the contragredient duality. The four main combinatorial classes are characterized in Lie-theoretic as well as homological terms. For the abelian category of nilpotent representations, we provide projective resolutions, standard homological invariants and explicit representation matrices of all indecomposables. Our approach can be extended to arrow ideal completions of path algebras of skew-gentle quivers.