Springer correspondence for the split symmetric pair in type A

Tsao Hsien Chen; Kari Vilonen; Ting Xue

doi:10.1112/S0010437X18007443

Springer correspondence for the split symmetric pair in type A

Tsao Hsien Chen, Kari Vilonen, Ting Xue

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Abstract

In this paper we establish Springer correspondence for the symmetric pair using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at . Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at arise in geometry.

Original language	English (US)
Pages (from-to)	2403-2425
Number of pages	23
Journal	Compositio Mathematica
Volume	154
Issue number	11
DOIs	https://doi.org/10.1112/S0010437X18007443
State	Published - Nov 1 2018
Externally published	Yes

Bibliographical note

Funding Information:
Chen Tsao-Hsien Vilonen Kari Xue Ting Department of Mathematics , University of Chicago , Chicago 60637 , USA email chenth@math.uchicago.edu School of Mathematics and Statistics , University of Melbourne , VIC 3010 , Australia email kari.vilonen@unimelb.edu.au Department of Mathematics and Statistics , University of Helsinki , Helsinki 00014 , Finland School of Mathematics and Statistics , University of Melbourne , VIC 3010 , Australia email ting.xue@unimelb.edu.au Department of Mathematics and Statistics , University of Helsinki , Helsinki 00014 , Finland The first author was supported in part by the AMS-Simons travel grant and the NSF grant DMS-1702337. The second author was supported in part by the ARC grants DP150103525 and DP180101445, the Academy of Finland, the Humboldt Foundation, the Simons Foundation, and the NSF grant DMS-1402928. The third author was supported in part by the ARC grants DE160100975, DP150103525 and the Academy of Finland. 11 10 2018 11 2018 154 11 2403 2425 28 01 2017 25 05 2018 © The Authors 2018 2018 The Authors In this paper we establish Springer correspondence for the symmetric pair $(\text{SL}(N),\text{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$ . Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry.

Publisher Copyright:
© 2018 The Author.

Keywords

Fourier transform
Hecke algebras
Hessenberg varieties
Springer correspondence
nearby cycle sheaves
symmetric pairs

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10.1112/S0010437X18007443

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abstract = "In this paper we establish Springer correspondence for the symmetric pair using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at . Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at arise in geometry.",

keywords = "Fourier transform, Hecke algebras, Hessenberg varieties, Springer correspondence, nearby cycle sheaves, symmetric pairs",

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note = "Funding Information: Chen Tsao-Hsien Vilonen Kari Xue Ting Department of Mathematics , University of Chicago , Chicago 60637 , USA email chenth@math.uchicago.edu School of Mathematics and Statistics , University of Melbourne , VIC 3010 , Australia email kari.vilonen@unimelb.edu.au Department of Mathematics and Statistics , University of Helsinki , Helsinki 00014 , Finland School of Mathematics and Statistics , University of Melbourne , VIC 3010 , Australia email ting.xue@unimelb.edu.au Department of Mathematics and Statistics , University of Helsinki , Helsinki 00014 , Finland The first author was supported in part by the AMS-Simons travel grant and the NSF grant DMS-1702337. The second author was supported in part by the ARC grants DP150103525 and DP180101445, the Academy of Finland, the Humboldt Foundation, the Simons Foundation, and the NSF grant DMS-1402928. The third author was supported in part by the ARC grants DE160100975, DP150103525 and the Academy of Finland. 11 10 2018 11 2018 154 11 2403 2425 28 01 2017 25 05 2018 {\textcopyright} The Authors 2018 2018 The Authors In this paper we establish Springer correspondence for the symmetric pair $(\text{SL}(N),\text{SO}(N))$ using Fourier transform, parabolic induction functor, and a nearby cycle sheaf construction. As an application of our results we see that the cohomology of Hessenberg varieties can be expressed in terms of irreducible representations of Hecke algebras of symmetric groups at $q=-1$ . Conversely, we see that the irreducible representations of Hecke algebras of symmetric groups at $q=-1$ arise in geometry. Publisher Copyright: {\textcopyright} 2018 The Author.",

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KW - Hessenberg varieties

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Springer correspondence for the split symmetric pair in type A

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