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 $L_{-1}(D_{\ell })$, $\ell \ge 5$. The block has exactly $\ell +1$ 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-$\mathcal{O}$ 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 $\hat{W}$-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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