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Mirabolic Satake equivalence and supergroups

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

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, ON M5S 2E4, Canada and Skolkovo Institute of Science and Technology, Moscow, Russia braval@math.toronto.edu
Michael Finkelberg
Affiliation:
Department of Mathematics, National Research University Higher School of Economics, Moscow 119048, Russia and Skolkovo Institute of Science and Technology, Moscow, Russia and Institute for the Information Transmission Problems, Moscow, Russia fnklberg@gmail.com
Victor Ginzburg
Affiliation:
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA vityaginzburg@gmail.com
Roman Travkin
Affiliation:
Skolkovo Institute of Science and Technology, Moscow 121205, Russia roman.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

Information

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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