Ramified tori and Gelfand--Kirillov dimension of mod p representations
Abstract
Let K/ℚp be a finite extension of degree n with p≥3, and let D be the non-split quaternion algebra over K. We prove that every infinite-dimensional admissible smooth representation of GL2(K) or D× with a central character and satisfying a Casimir annihilation condition has Gelfand--Kirillov dimension n. As a corollary, every infinite-dimensional admissible Banach representation whose locally analytic vectors have an infinitesimal character has dimension n, strengthening Sorgdrager's bound to an equality. An appendix by Yongquan Hu proves that, when K≠ℚp and the coefficient field is algebraically closed, the canonical diagram of an admissible supersingular representation satisfying the same Casimir annihilation condition is (π,π,id), as in the case of local fields of characteristic p.
Related
- Gelfand--Kirillov dimensions of highest weight modules for basic classical Lie superalgebras
- Jacobson radical non-nil algebras of Gel'fand-Kirillov dimension 2
- Gelfand--Tsetlin bases for representations of Uq(gln) at roots of unity
- Non-admissible smooth irreducible representations of GL2(F) for totally ramified extensions F/ℚp
- Galois representations ramified at one prime via relative deformation theory
BibTeX
@misc{zhao2026ramified,
title = {{Ramified tori and Gelfand--Kirillov dimension of mod $p$ representations}},
author = {Jiusi Zhao and Yongquan Hu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04350},
eprint = {2611.04350},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04350},
}
RIS
TY - UNPB
TI - Ramified tori and Gelfand--Kirillov dimension of mod p representations
AU - Jiusi Zhao
AU - Yongquan Hu
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04350
UR - https://arxiv.org/abs/2611.04350
AB - Let K/ℚ_p be a finite extension of degree n with p≥3, and let D be the non-split quaternion algebra over K. We prove that every infinite-dimensional admissible smooth representation of GL_2(K) or D^× with a central character and satisfying a Casimir annihilation condition has Gelfand--Kirillov dimension n. As a corollary, every infinite-dimensional admissible Banach representation whose locally analytic vectors have an infinitesimal character has dimension n, strengthening Sorgdrager's bound to an equality. An appendix by Yongquan Hu proves that, when K≠ℚ_p and the coefficient field is algebraically closed, the canonical diagram of an admissible supersingular representation satisfying the same Casimir annihilation condition is (π,π,id), as in the case of local fields of characteristic p.
ER -