Coherent Springer theory and the categorical Deligne-Langlands correspondence
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arXiv
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| Format: | Preprint |
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2020
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| author | Ben-Zvi, David Chen, Harrison Helm, David Nadler, David |
| author_facet | Ben-Zvi, David Chen, Harrison Helm, David Nadler, David |
| contents | Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an equivariant $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields $F$ with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from $K$-theory to Hochschild homology and thereby identify $\mathcal{H}$ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of $\mathcal{H}$-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu).
In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of $\mathrm{GL}_n(F)$ into coherent sheaves on the stack of Langlands parameters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_02321 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Coherent Springer theory and the categorical Deligne-Langlands correspondence Ben-Zvi, David Chen, Harrison Helm, David Nadler, David Representation Theory Algebraic Geometry Number Theory Kazhdan and Lusztig identified the affine Hecke algebra $\mathcal{H}$ with an equivariant $K$-group of the Steinberg variety, and applied this to prove the Deligne-Langlands conjecture, i.e., the local Langlands parametrization of irreducible representations of reductive groups over nonarchimedean local fields $F$ with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from $K$-theory to Hochschild homology and thereby identify $\mathcal{H}$ with the endomorphisms of a coherent sheaf on the stack of unipotent Langlands parameters, the coherent Springer sheaf. As a result the derived category of $\mathcal{H}$-modules is realized as a full subcategory of coherent sheaves on this stack, confirming expectations from strong forms of the local Langlands correspondence (including recent conjectures of Fargues-Scholze, Hellmann and Zhu). In the case of the general linear group our result allows us to lift the local Langlands classification of irreducible representations to a categorical statement: we construct a full embedding of the derived category of smooth representations of $\mathrm{GL}_n(F)$ into coherent sheaves on the stack of Langlands parameters. |
| title | Coherent Springer theory and the categorical Deligne-Langlands correspondence |
| topic | Representation Theory Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2010.02321 |