Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical Groups
Abstract
Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical GroupsLet q be a power of an odd prime. For the unipotent radicals of maximal rational parabolic subgroups of the finite symplectic, orthogonal and unitary groups, we determine all complex irreducible characters, their values, and the number of characters of each degree. Each of these numbers is a polynomial in q−1 with nonnegative integer coefficients, in analogy with a conjecture of Isaacs for unitriangular groups. For the orthogonal radicals, the character table and the power maps do not depend on the Witt type of the residual form, whereas the isomorphism type of the radical does when the isotropic dimension is at least three and the residual dimension is positive and even. Consequently, for 3≤ a≤ n−1 the radicals of the stabilizers of totally isotropic a-spaces in SO+2n(q) and SO-2n(q) form Brauer pairs; for a≤ n−2 both stabilizers are maximal parabolic subgroups. Finally, we express the multiplicities through R-polynomials, with the unequal parameters q2,…,q2,q in the unitary case.
Related
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BibTeX
@misc{hao2026characters,
title = {{Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical Groups}},
author = {Qingchun Hao and Yang Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29976},
eprint = {2610.29976},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29976},
}
RIS
TY - UNPB
TI - Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical Groups
AU - Qingchun Hao
AU - Yang Yang
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29976
UR - https://arxiv.org/abs/2610.29976
AB - Characters and Polynomiality for Maximal Parabolic Unipotent Radicals in Finite Classical GroupsLet q be a power of an odd prime. For the unipotent radicals of maximal rational parabolic subgroups of the finite symplectic, orthogonal and unitary groups, we determine all complex irreducible characters, their values, and the number of characters of each degree. Each of these numbers is a polynomial in q−1 with nonnegative integer coefficients, in analogy with a conjecture of Isaacs for unitriangular groups. For the orthogonal radicals, the character table and the power maps do not depend on the Witt type of the residual form, whereas the isomorphism type of the radical does when the isotropic dimension is at least three and the residual dimension is positive and even. Consequently, for 3≤ a≤ n−1 the radicals of the stabilizers of totally isotropic a-spaces in SO^+_2n(q) and SO^-_2n(q) form Brauer pairs; for a≤ n−2 both stabilizers are maximal parabolic subgroups. Finally, we express the multiplicities through R-polynomials, with the unequal parameters q^2,…,q^2,q in the unitary case.
ER -