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Parabolic induction for Takiff algebras and character formulas in Chari's category π’ͺΜƒ

  • Fulin Chen
  • Xin Huang
  • Ruling Zhou
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Abstract

Let g be a finite-dimensional complex semisimple Lie algebra, and let π’žNg=gβŠ— (β„‚[t]/tN+1β„‚[t]) denote the N-th Takiff algebra of g. We introduce a notion of parabolic subalgebras of π’žNg and classify them in terms of (N+2)-partitions of a simple root system of g. We then study the corresponding parabolically induced π’žNg-modules and establish a necessary and sufficient criterion for their irreducibility, extending Jantzen's criterion for parabolically induced g-modules. As an application, we derive character formulas for almost all irreducible Harish-Chandra modules in Chari's category π’ͺΜƒ over the (centerless) affine Lie algebra associated with g. In particular, when g=sl3(β„‚), we obtain explicit character formulas for all irreducible Harish-Chandra modules in this category.

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Cite

BibTeXRIS
BibTeX
@misc{chen2026parabolic,
  title = {{Parabolic induction for Takiff algebras and character formulas in Chari's category $\widetilde\mathcalO$}},
  author = {Fulin Chen and Xin Huang and Ruling Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.02787},
  eprint = {2610.02787},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.02787},
}
RIS
TY  - UNPB
TI  - Parabolic induction for Takiff algebras and character formulas in Chari's category π’ͺΜƒ
AU  - Fulin Chen
AU  - Xin Huang
AU  - Ruling Zhou
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.02787
UR  - https://arxiv.org/abs/2610.02787
AB  - Let g be a finite-dimensional complex semisimple Lie algebra, and let π’ž_Ng=gβŠ— (β„‚[t]/t^N+1β„‚[t]) denote the N-th Takiff algebra of g. We introduce a notion of parabolic subalgebras of π’ž_Ng and classify them in terms of (N+2)-partitions of a simple root system of g. We then study the corresponding parabolically induced π’ž_Ng-modules and establish a necessary and sufficient criterion for their irreducibility, extending Jantzen's criterion for parabolically induced g-modules. As an application, we derive character formulas for almost all irreducible Harish-Chandra modules in Chari's category π’ͺΜƒ over the (centerless) affine Lie algebra associated with g. In particular, when g=sl_3(β„‚), we obtain explicit character formulas for all irreducible Harish-Chandra modules in this category.
ER  -