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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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Picard sheaves, local Brauer groups, and topological modular forms
Abstract We develop tools to analyze and compare the Brauer groups of spectra such as periodic complex and real ‐theory and topological modular forms, as well as the derived moduli stack of elliptic curves. In particular, we prove that the Brauer group of is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Our main computational focus is on the subgroup of the Brauer group consisting of elements trivialized by some étale extension, which we call the local Brauer group. Essential information about this group can be accessed by a thorough understanding of the Picard sheaf and its cohomology. We deduce enough information about the Picard sheaf of and the (derived) moduli stack of elliptic curves to determine the structure of their local Brauer groups away from the prime 2. At 2, we show that they are both infinitely generated and agree up to a potential error term that is a finite 2‐torsion group.
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- Award ID(s):
- 2102010
- PAR ID:
- 10508811
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Journal of Topology
- Volume:
- 17
- Issue:
- 2
- ISSN:
- 1753-8416
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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