arXiv 2102.112472021-02math.RT

广义半无限Hecke等价与局部几何Langlands对应

A generalized semi-infinite Hecke equivalence and the local geometric Langlands correspondence

Alexey Sevostyanov

备注：: A crucial mistake has been discovered in the proof of the main statement

摘要：

我们引入了一类等价，我们称之为广义半无限Hecke等价，某些类别的表示的分次结合代数之间出现在设置的半无限上同调的结合代数和类别的表示的相关代数的Hecke型，我们称之为半无限Hecke代数。作为应用，我们得到了水平$-2h^\vee-k$的非扭仿射李代数$\widehat{\mathfrak g}$的一类表示与水平$k$的$\widehat{\mathfrak g}$的W-代数的双生成表示的等价性，其中$h^\vee$是基础半单李代数$\mathfrak g$和$k\in \mathbb{C}$的对偶Coxeter数.当$k=-h^\vee$时，这产生中心电荷$-h^\vee$的$\widehat{\mathfrak g}$的表示范畴与穿孔圆盘$D^\times$上的$^LG$ -pers的空间${\rm Op}_{^LG}(D^\times)$上的相干层范畴${\rm Coh}({\rm Op}_{^LG}(D^\times))$之间的等价，其中$^LG$是与李代数$\mathfrak g$伴随型代数群的朗兰兹对偶群。这可以看作是局部几何朗兰兹对应的一个版本。上述等价推广到仿射李代数的情形，推广了$\mathfrak g$的广义Gelfand-Graev表示范畴与相应的广义生成W-代数的表示范畴之间的Skryabin等价，以及Kostant关于$\mathfrak g$上Whittaker模分类的结果.

英文摘要：

We introduce a class of equivalences, which we call generalized semi-infinite Hecke equivalences, between certain categories of representations of graded associative algebras which appear in the setting of semi-infinite cohomology for associative algebras and categories of representations of related algebras of Hecke type which we call semi-infinite Hecke algebras. As an application we obtain an equivalence between a category of representations of a non-twisted affine Lie algebra $\widehat{\mathfrak g}$ of level $-2h^\vee-k$, where $h^\vee$ is the dual Coxeter number of the underlying semisimple Lie algebra $\mathfrak g$ and $k\in \mathbb{C}$, and the category of finitely generated representations of the W-algebra associated to $\widehat{\mathfrak g}$ of level $k$. When $k=-h^\vee$ this yields an equivalence between a category of representations of $\widehat{\mathfrak g}$ of central charge $-h^\vee$ and the category ${\rm Coh}({\rm Op}_{^LG}(D^\times))$ of coherent sheaves on the space ${\rm Op}_{^LG}(D^\times)$ of $^LG$-opers on the punctured disc $D^\times$, where $^LG$ is the Langlands dual group to the algebraic group of adjoint type with Lie algebra $\mathfrak g$. This can be regarded as a version of the local geometric Langlands correspondence. The above mentioned equivalences generalize to the case of affine Lie algebras the Skryabin equivalence between the categories of generalized Gelfand-Graev representations of $\mathfrak g$ and the categories of representations of the corresponding finitely generated W-algebras, and Kostant's results on the classification of Whittaker modules over $\mathfrak g$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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