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- Title
Coherent Springer theory and the categorical Deligne-Langlands correspondence.
- Authors
Ben-Zvi, David; Chen, Harrison; Helm, David; Nadler, David
- Abstract
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.
- Subjects
SHEAF theory; ALGEBRAIC geometry; HECKE algebras; ENDOMORPHISMS; REPRESENTATIONS of groups (Algebra); LOGICAL prediction
- Publication
Inventiones Mathematicae, 2024, Vol 235, Issue 2, p255
- ISSN
0020-9910
- Publication type
Article
- DOI
10.1007/s00222-023-01224-2