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Springer correspondence for the split symmetric pair in type $A$

Part of: Algebraic groups Linear algebraic groups and related topics

Published online by Cambridge University Press:  11 October 2018

Tsao-Hsien Chen ,
Kari Vilonen  and
Ting Xue
Tsao-Hsien Chen
Affiliation:
Department of Mathematics, University of Chicago, Chicago 60637, USA email chenth@math.uchicago.edu
Kari Vilonen
Affiliation:
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
Ting Xue
Affiliation:
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
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Abstract

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.

Keywords

Springer correspondencenearby cycle sheavessymmetric pairsHecke algebrasHessenberg varietiesFourier transform

MSC classification

Primary: 20G99: None of the above, but in this section 14L99: None of the above, but in this section

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 11 , November 2018 , pp. 2403 - 2425
Copyright
© The Authors 2018 

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Footnotes

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.

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