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A generalized semi-infinite Hecke equivalence and the local geometric Langlands correspondence
Representation Theory · Submitted 22 Feb 2021
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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 $\mathfrak g$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 1.00 | 0.95 | 0.36 | asserted 1.00 |
| s1 | 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 $k$. | result 0.98 | 0.93 | 0.33 | asserted 1.00 |
| s2 | 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 $\mathfrak g$. | result 0.98 | 0.87 | 0.26 | asserted 1.00 |
| s3 | This can be regarded as a version of the local geometric Langlands correspondence. | interpretation 1.00 | 0.53 | 0.20 | hedged possibility 0.62 |
| s4 | 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 $\mathfrak g$. | interpretation 0.74 | 0.67 | 0.09 | asserted 0.97 |
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Original abstract
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 $\mathfrak g$.
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