Grothendieck Group of the Level -1 Type D: Gaps in Jin's Proof and a Complete Proof

Abstract

Jin recently claims that the distinguished level $-1$ vacuum block of $L_{-1}(D_\ell)$ realizes the specialized dual affine left-cell module attached to the subregular cell containing $s_0$. We show that the proof of this Grothendieck-group statement contains a gap. We give an explicit counterexample over the discrete valuation ring $\Bbbk[[t]]$, showing that the categorical implication used in the proof is false in general. Consequently, Jin's Theorem~6.3 does not follow from the argument given in his Section~6.2. We then supply a complete replacement proof for the grading-restricted level-$-1$ vacuum block. Combining the classification of simple objects, we prove finite length, construct an injective signed normalized-character map into a completed singular orbit module, and identify its image with the specialized dual subregular left-cell module. Thus the cell-module realization claimed is established for the grading-restricted vacuum block.