Abstract
To any element of a connected, simply connected, semisimple complex algebraic group đș and a choice of an element of the corresponding Weyl group, there is an associated Lusztig variety. When the element of đș 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 KazhdanâLusztig basis of the corresponding Hecke algebra. In type đŽ, we prove a more precise statement: that the Frobenius character of this action is precisely the symmetric function given by the characters of a KazhdanâLusztig basis element. The main idea is to find cellular decompositions of desingularizations of these varieties and apply the BrosnanâChow 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 BrosnanâChow [P. Brosnan and T.âY. Chow, Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties, Adv. Math. 329 (2018), 955â1001] solution to the ShareshianâWachs [J. Shareshian and M.âL. Wachs, Chromatic quasisymmetric functions, Adv. Math. 295 (2016), 497â551] 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.
References
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© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive bi-Ricci curvature
- Ceresa cycles of bielliptic Picard curves
- A geometric approach to characters of Hecke algebras
- Non-commutative nature of â-adic vanishing cycles
- Quadratically enriched tropical intersections
- Approximation of maps between real algebraic varieties
- Towards a~better understanding of đđ functions and definable đđ functions of several variables
- Every salami has two ends
Articles in the same Issue
- Frontmatter
- Optimal volume bound and volume growth for Ricci-nonnegative manifolds with positive bi-Ricci curvature
- Ceresa cycles of bielliptic Picard curves
- A geometric approach to characters of Hecke algebras
- Non-commutative nature of â-adic vanishing cycles
- Quadratically enriched tropical intersections
- Approximation of maps between real algebraic varieties
- Towards a~better understanding of đđ functions and definable đđ functions of several variables
- Every salami has two ends