Representation theory of the Gelfand quiver and Harish-Chandra modules for $\mathsf{SL}_2(\mathbb{R})$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912993601650688 |
|---|---|
| author | Burban, Igor Gnedin, Wassilij |
| author_facet | Burban, Igor Gnedin, Wassilij |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00274 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Representation theory of the Gelfand quiver and Harish-Chandra modules for $\mathsf{SL}_2(\mathbb{R})$ Burban, Igor Gnedin, Wassilij Representation Theory 16Gxx (Primary) 22E46, 16E35 (Secondary) 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. |
| title | Representation theory of the Gelfand quiver and Harish-Chandra modules for $\mathsf{SL}_2(\mathbb{R})$ |
| topic | Representation Theory 16Gxx (Primary) 22E46, 16E35 (Secondary) |
| url | https://arxiv.org/abs/2604.00274 |