Subregular affine cells and the level −1 vertex algebra of type D
Abstract
We prove the simple-object prediction of Shan--Yan--Zhao and a basis-preserving dual-cell realization for the distinguished vacuum block of the simple affine vertex algebras L−1(Dℓ), ℓ≥5. The block has exactly ℓ+1 simple objects, indexed by the subregular affine left cell containing s0. The proof combines a primitive-ideal inclusion, an independent exhaustion argument, and a finite-length step. A noncritical Sugawara lift supplies finite-dimensional weight-space detectors in the original Shan--Yan--Zhao category-𝒪 block, so dévissage applies to its ordinary Grothendieck group. We then identify this group, basis by basis, with the q=1 specialization of the corresponding dual affine left-cell module. An injective signed normalized-character realization identifies the resulting image with the canonical dual-cell image in the completed singular-orbit module and hence supplies the corresponding abstract Ŵ-module structure. We do not identify this action with a functorial action arising from affine twisting functors or Kashiwara--Tanisaki localization. The subregular inverse Kazhdan--Lusztig calculation of Bezrukavnikov--Kac--Krylov also yields uniform character formulas.
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BibTeX
@misc{jin2026subregular,
title = {{Subregular affine cells and the level $-1$ vertex algebra of type $D$}},
author = {Sihai Jin},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.11997},
eprint = {2608.11997},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11997},
}
RIS
TY - UNPB
TI - Subregular affine cells and the level −1 vertex algebra of type D
AU - Sihai Jin
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.11997
UR - https://arxiv.org/abs/2608.11997
AB - We prove the simple-object prediction of Shan--Yan--Zhao and a basis-preserving dual-cell realization for the distinguished vacuum block of the simple affine vertex algebras L_−1(D_ℓ), ℓ≥5. The block has exactly ℓ+1 simple objects, indexed by the subregular affine left cell containing s_0. The proof combines a primitive-ideal inclusion, an independent exhaustion argument, and a finite-length step. A noncritical Sugawara lift supplies finite-dimensional weight-space detectors in the original Shan--Yan--Zhao category-𝒪 block, so dévissage applies to its ordinary Grothendieck group. We then identify this group, basis by basis, with the q=1 specialization of the corresponding dual affine left-cell module. An injective signed normalized-character realization identifies the resulting image with the canonical dual-cell image in the completed singular-orbit module and hence supplies the corresponding abstract Ŵ-module structure. We do not identify this action with a functorial action arising from affine twisting functors or Kashiwara--Tanisaki localization. The subregular inverse Kazhdan--Lusztig calculation of Bezrukavnikov--Kac--Krylov also yields uniform character formulas.
ER -