Non-canonical Oscillator Representations of Orthogonal Lie Algebras and Pfaffian Varieties
Abstract
Associated varieties are fundamental geometric invariants in the study of infinite-dimensional representations of semisimple Lie algebras. Luo and Xu (2013) constructed a large family of infinite-dimensional irreducible representations of orthogonal Lie algebras by applying Fourier transforms to the natural oscillator representations. In this paper, we prove that the associated varieties of these irreducible representations are isomorphic to explicitly described coordinate linear sections of Pfaffian varieties. The modules are realized as spaces of polynomial solutions of transformed Laplace equations. We construct filtrations from weighted polynomial degrees and determine the annihilator ideals of the corresponding associated graded modules. The defining equations consist of linear coordinate equations, minors and quadratic Pfaffian relations. The even and odd orthogonal Lie algebras are treated separately. The associated varieties are viewed as reduced affine schemes over a field of characteristic zero. In each irreducible family with fixed Fourier parameters, the associated variety is independent of the grading parameter. When the two Fourier parameters coincide, the projectivizations in the even case are Grassmannians.
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BibTeX
@misc{zhang2026noncanonical,
title = {{Non-canonical Oscillator Representations of Orthogonal Lie Algebras and Pfaffian Varieties}},
author = {Hengjia Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25157},
eprint = {2610.25157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25157},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Non-canonical Oscillator Representations of Orthogonal Lie Algebras and Pfaffian Varieties
AU - Hengjia Zhang
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25157
UR - https://arxiv.org/abs/2610.25157
AB - Associated varieties are fundamental geometric invariants in the study of infinite-dimensional representations of semisimple Lie algebras. Luo and Xu (2013) constructed a large family of infinite-dimensional irreducible representations of orthogonal Lie algebras by applying Fourier transforms to the natural oscillator representations. In this paper, we prove that the associated varieties of these irreducible representations are isomorphic to explicitly described coordinate linear sections of Pfaffian varieties. The modules are realized as spaces of polynomial solutions of transformed Laplace equations. We construct filtrations from weighted polynomial degrees and determine the annihilator ideals of the corresponding associated graded modules. The defining equations consist of linear coordinate equations, minors and quadratic Pfaffian relations. The even and odd orthogonal Lie algebras are treated separately. The associated varieties are viewed as reduced affine schemes over a field of characteristic zero. In each irreducible family with fixed Fourier parameters, the associated variety is independent of the grading parameter. When the two Fourier parameters coincide, the projectivizations in the even case are Grassmannians.
ER -