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Smith theory and geometric Hecke algebras

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Abstract

In 1960 Borel proved a “localization” result relating the rational cohomology of a topological space X to the rational cohomology of the fixed points for a torus action on X. This result and its generalizations have many applications in Lie theory. In 1934, Smith proved a similar localization result relating the mod p cohomology of X to the mod p cohomology of the fixed points for a \({\mathbb {Z}}/p\)-action on X. In this paper we study \({\mathbb {Z}}/p\)-localization for constructible sheaves and functions. We show that \({\mathbb {Z}}/p\)-localization on loop groups is related via the geometric Satake correspondence to some special homomorphisms that exist between algebraic groups defined over a field of small characteristic.

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Acknowledgements

I thank Florian Herzig, Gopal Prasad, and Ting Xue for help with algebraic groups. In particular, most of the material in Sect. 3.4.5 I learned from Ting. While developing these ideas I benefited from discussions with Paul Goerss, David Nadler, and Zhiwei Yun. I thank Sam Evens for correcting some microlocal mistakes in an earlier version of this paper, and Akshay Venkatesh for improving the proof of Theorem 3.3.

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  1. Boston College, Chestnut Hill, MA, USA

    David Treumann

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  1. David Treumann
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Treumann, D. Smith theory and geometric Hecke algebras. Math. Ann. 375, 595–628 (2019). https://doi.org/10.1007/s00208-019-01860-1

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