Bibliographic Information: Nie, S. (2024). Decomposition of higher Deligne-Lusztig representations. arXiv preprint arXiv:2406.06430v3.
Research Objective: The paper aims to decompose elliptic higher Deligne-Lusztig representations into irreducible components and explore their connection to Yu's representations, which are known to induce irreducible supercuspidal representations.
Methodology: The author introduces a new variety, Zφ,U,r, with a simpler structure than the higher Deligne-Lusztig variety. By proving an equality between representations associated with both varieties, the focus shifts to decomposing the representation associated with Zφ,U,r. This is achieved by leveraging Howe factorization of smooth characters and proving a concentration-at-one-degree property for Zφ,U,r.
Key Findings:
Main Conclusions: The study reveals a deep connection between higher Deligne-Lusztig representations and supercuspidal representations, advancing the understanding of both. The explicit decomposition offers a new perspective on the structure of these representations and their role in the representation theory of p-adic groups.
Significance: This work significantly contributes to the field of representation theory, particularly in the context of Deligne-Lusztig theory and its applications to the representation theory of p-adic groups. The findings provide a new framework for understanding the structure and properties of higher Deligne-Lusztig representations and their relationship to supercuspidal representations.
Limitations and Future Research: The paper primarily focuses on unramified cuspidal G-data. Further research could explore the decomposition and properties of higher Deligne-Lusztig representations associated with more general cuspidal data. Additionally, investigating the precise relationship between the Weil-Heisenberg representations and their geometric analogs, as highlighted in the paper, could be a fruitful avenue for future work.
To Another Language
from source content
arxiv.org
Deeper Inquiries