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Tilting sheaves for real groups and Koszul duality

Part of: Lie groups Special varieties

Published online by Cambridge University Press:  11 September 2025

Andrei Ionov Open the ORCID record for Andrei Ionov [Opens in a new window]  and
Zhiwei Yun
Andrei Ionov
Affiliation:
The University of Texas at Austin Department of Mathematics - PMA 8.100, 2515 Speedway, Stop C1200, Austin, TX 78712, USA andrei.ionov@austin.utexas.edu1
Zhiwei Yun
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA zyun@mit.edu1
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Abstract

For a certain class of real analytic varieties with Lie group actions, we develop a theory of (free-monodromic) tilting sheaves, and apply it to flag varieties stratified by real group orbits. For quasi-split real groups, we construct a fully faithful embedding of the category of tilting sheaves to a real analog of the category of Soergel bimodules, establishing real group analogs of Soergel’s structure theorem and the endomorphism theorem. We apply these results to give a purely geometric proof of the main result of Bezrukavnikov and Vilonen [Koszul duality for quasi-split real groups, Invent. Math. 226 (2021), 139–193], which proves Soergel’s conjecture [Langlands’ philosophy and Koszul duality, in Algebra – representation theory (Constanta, 2000), NATO Science Series II: Mathematics, Physics and Chemistry, vol. 28 (Kluwer Academic Publishers, Dordrecht, 2001), 379–414] for quasi-split groups.

Keywords

real algebraic groupstilting sheavesKoszul dualitySoergel modules

MSC classification

Primary: 22E47: Representations of Lie and real algebraic groups 14P25: Topology of real algebraic varieties 14M15: Grassmannians, Schubert varieties, flag manifolds

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 7 , July 2025 , pp. 1698 - 1777
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Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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