Level -1型D的Grothendieck群：Jin证明中的漏洞与完整证明
Grothendieck Group of the Level -1 Type D: Gaps in Jin's Proof and a Complete Proof
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中文总结 AI 辅助
针对Jin关于$L_{-1}(D_\ell)$的level -1真空块实现次正则左胞腔模的结论，指出其Grothendieck群证明存在漏洞并给出反例，同时为分次限制真空块的该结论提供了完整证明。
中文摘要 AI 辅助
Jin近期声称，$L_{-1}(D_\ell)$的特异level $-1$真空块实现了与包含$s_0$的次正则胞腔对应的特殊化对偶仿射左胞腔模。我们指出这一Grothendieck群命题的证明存在一处漏洞。我们在离散赋值环$\Bbbk[[t]]$上构造了一个明确的反例，表明证明中使用的范畴蕴含关系一般不成立。因此，Jin的定理6.3无法由其第6.2节给出的论证推导得出。\n随后我们为分次限制的level-$-1$真空块提供了完整的替代证明。结合单对象的分类，我们证明了有限长度性质，构造了一个到完备奇异轨道模的带符号归一化特征单射，并将其像识别为特殊化对偶次正则左胞腔模。由此，所声称的胞腔模实现对于分次限制真空块得以确立。
英文摘要
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.
