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Mirabolic Satake equivalence and supergroups

Part of: Special varieties Lie algebras and Lie superalgebras Families, fibrations

Published online by Cambridge University Press:  22 July 2021

Alexander Braverman ,
Michael Finkelberg ,
Victor Ginzburg  and
Roman Travkin
Alexander Braverman
Affiliation:
Department of Mathematics, University of Toronto, Toronto, ONM5S 2E4, Canada and Skolkovo Institute of Science and Technology, Moscow, Russiabraval@math.toronto.edu
Michael Finkelberg
Affiliation:
Department of Mathematics, National Research University Higher School of Economics, Moscow119048, Russia and Skolkovo Institute of Science and Technology, Moscow, Russia and Institute for the Information Transmission Problems, Moscow, Russiafnklberg@gmail.com
Victor Ginzburg
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL60637, USAvityaginzburg@gmail.com
Roman Travkin
Affiliation:
Skolkovo Institute of Science and Technology, Moscow121205, Russiaroman.travkin2012@gmail.com
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Abstract

We construct a mirabolic analogue of the geometric Satake equivalence. We also prove an equivalence that relates representations of a supergroup to the category of $\operatorname{GL}(N-1,{\mathbb {C}}[\![t]\!])$-equivariant perverse sheaves on the affine Grassmannian of $\operatorname{GL}_N$. We explain how our equivalences fit into a more general framework of conjectures due to Gaiotto and to Ben-Zvi, Sakellaridis and Venkatesh.

Keywords

Satake equivalencemirabolic affine Grassmanniansupergroups

MSC classification

Secondary: 14D24: Geometric Langlands program 14M15: Grassmannians, Schubert varieties, flag manifolds 17B20: Simple, semisimple, reductive (super)algebras
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 8 , August 2021 , pp. 1724 - 1765
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Copyright
© 2021 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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Footnotes

To our friend Sasha Shen on the occasion of his 60th birthday

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