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A geometric approach to characters of Hecke algebras

  • Alex Abreu EMAIL logo and Antonio Nigro
Published/Copyright: January 18, 2025

Abstract

To any element of a connected, simply connected, semisimple complex algebraic group đș and a choice of an element of the corresponding Weyl group, there is an associated Lusztig variety. When the element of đș is regular semisimple, the corresponding variety carries an action of the Weyl group on its (equivariant) intersection cohomology. From this action, we recover the induced characters of an element of the Kazhdan–Lusztig basis of the corresponding Hecke algebra. In type 𝐮, we prove a more precise statement: that the Frobenius character of this action is precisely the symmetric function given by the characters of a Kazhdan–Lusztig basis element. The main idea is to find cellular decompositions of desingularizations of these varieties and apply the Brosnan–Chow palindromicity criterion for determining when the local invariant cycle map is an isomorphism. This recovers some results of Lusztig about character sheaves and gives a generalization of the Brosnan–Chow [P. Brosnan and T. Y. Chow, Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties, Adv. Math. 329 (2018), 955–1001] solution to the Shareshian–Wachs [J. Shareshian and M. L. Wachs, Chromatic quasisymmetric functions, Adv. Math. 295 (2016), 497–551] conjecture to non-codominant permutations, where singularities are involved.

Acknowledgements

We would like to thank Arun Ram for pointing out to us the reference [36]. We also thank the referee for carefully reading the paper.

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Received: 2023-02-27
Revised: 2024-12-11
Published Online: 2025-01-18
Published in Print: 2025-04-01

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