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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, thecoherent 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$$\mathrm{GL}_{n}(F)$$ into coherent sheaves on the stack of Langlands parameters.
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The trace of the affine Hecke category
Abstract We compare the (horizontal) trace of the affine Hecke category with the elliptic Hall algebra, thus obtaining an “affine” version of the construction of Gorsky et al. (Int. Math. Res. Not. IMRN2022(2022) 11304–11400). Explicitly, we show that the aforementioned trace is generated by the objects as , where denote the Wakimoto objects of Elias and denote Rouquier complexes. We compute certain categorical commutators between the 's and show that they match the categorical commutators between the sheaves on the flag commuting stack that were considered in Neguț (Publ. Math. Inst. Hautes Études Sci. 135 (2022) 337–418). At the level of ‐theory, these commutators yield a certain integral form of the elliptic Hall algebra, which we can thus map to the ‐theory of the trace of the affine Hecke category.
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- Award ID(s):
- 1760329
- PAR ID:
- 10420293
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Proceedings of the London Mathematical Society
- Volume:
- 126
- Issue:
- 6
- ISSN:
- 0024-6115
- Page Range / eLocation ID:
- p. 2013-2056
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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