---
_id: '66293'
abstract:
- lang: eng
  text: 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.
author:
- first_name: Igor
  full_name: Burban, Igor
  id: '72064'
  last_name: Burban
- first_name: Wassilij
  full_name: Gnedin, Wassilij
  last_name: Gnedin
citation:
  ama: Burban I, Gnedin W. Representation theory of the Gelfand quiver and Harish-Chandra
    modules for SL_2(R). <i>arXiv:260400274</i>. Published online 2026.
  apa: Burban, I., &#38; Gnedin, W. (2026). Representation theory of the Gelfand quiver
    and Harish-Chandra modules for SL_2(R). In <i>arXiv:2604.00274</i>.
  bibtex: '@article{Burban_Gnedin_2026, title={Representation theory of the Gelfand
    quiver and Harish-Chandra modules for SL_2(R)}, journal={arXiv:2604.00274}, author={Burban,
    Igor and Gnedin, Wassilij}, year={2026} }'
  chicago: Burban, Igor, and Wassilij Gnedin. “Representation Theory of the Gelfand
    Quiver and Harish-Chandra Modules for SL_2(R).” <i>ArXiv:2604.00274</i>, 2026.
  ieee: I. Burban and W. Gnedin, “Representation theory of the Gelfand quiver and
    Harish-Chandra modules for SL_2(R),” <i>arXiv:2604.00274</i>. 2026.
  mla: Burban, Igor, and Wassilij Gnedin. “Representation Theory of the Gelfand Quiver
    and Harish-Chandra Modules for SL_2(R).” <i>ArXiv:2604.00274</i>, 2026.
  short: I. Burban, W. Gnedin, ArXiv:2604.00274 (2026).
date_created: 2026-07-07T06:28:32Z
date_updated: 2026-07-07T06:29:40Z
external_id:
  arxiv:
  - '2604.00274'
language:
- iso: eng
publication: arXiv:2604.00274
status: public
title: Representation theory of the Gelfand quiver and Harish-Chandra modules for
  SL_2(R)
type: preprint
user_id: '72064'
year: '2026'
...