Abstract
We give geometric descriptions of the category \(C_k(n,d)\) of rational polynomial representations of \(GL_n\) over a field \(k\) of degree \(d\) for \(d\le n\), the Schur functor and Schur–Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category \(C_k(n,d)\) and the Schur functor.
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Notes
While Mirković works with \({\mathcal {D}}\)-modules, the same argument works in the constructible context with arbitrary coefficients.
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Acknowledgments
This paper has been a long time coming and so the author had a number of years to benefit from useful conversations and deep insights from many people. He would like to thank in particular: David Ben-Zvi for continued support and advice, Daniel Juteau whose thesis was a source of inspiration for much of this paper, Pramod Achar for encouragement, Geordie Williamson for comments on a draft of the paper and an anonymous referee for a careful reading and many helpful comments. Thanks as well to Dennis Gaitsgory, Joel Kamnitzer, David Helm, David Nadler, Catharina Stroppel, Zhiwei Yun, and Xinwen Zhu.
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The author was supported by an NSF postdoctoral research fellowship.
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Mautner, C. A geometric Schur functor. Sel. Math. New Ser. 20, 961–977 (2014). https://doi.org/10.1007/s00029-014-0147-9
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DOI: https://doi.org/10.1007/s00029-014-0147-9