Coherent Springer theory and the categorical Deligne-Langlands correspondence

• 成果类型: Article
• 署名作者: Ben-Zvi, David; Chen, Harrison; Helm, David; Nadler, David
• 署名单位: University of Texas System; University of Texas Austin; Academia Sinica - Taiwan; Imperial College London; University of California System; University of California Berkeley
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-023-01224-2
• 发表日期: 2024
• 页码: 255-344
• 关键词: integral-transforms; deformation rings; REPRESENTATIONS; Algebra; sheaves; traces

摘要

Kazhdan and Lusztig identified the affine Hecke algebra 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 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 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 GL(n)(F) into coherent sheaves on the stack of Langlands parameters.