Coherent Springer theory and the categorical Deligne-Langlands correspondence
arXiv:2010.02321 · doi:10.1007/s00222-023-01224-2
Abstract
Kazhdan and Lusztig identified the affine Hecke algebra with an equivariant -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 with an Iwahori-fixed vector. We apply techniques from derived algebraic geometry to pass from -theory to Hochschild homology and thereby identify 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 -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 into coherent sheaves on the stack of Langlands parameters.
65 pages. Published version
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