PreprintarXiv licencearXiv
Parabolic filtrations of Soergel bimodules
Abstract
Given a Coxeter system (W,S) and a subset I⊂ S. Let WI=⟨ I⟩ be the standard parabolic subgroup of a Coxeter group W, and IW be the set of representatives of minimal length in the coset WIbackslash W. We construct a parabolic version of Soergel's standard filtration, and show that, given any Soergel bimodule, each subquotient of its parabolic filtration is a direct sum of grading shifts of BxI⊗R Ry with x∈ WI and y∈ IW. By applying the character theorem of Elias and Williamson, we deduce the positivity of coefficients in the expansion of Cw with respect to the hybrid basis {CxTy:x∈ WI,y∈ IW}. This generalizes Grojnowski and Haiman's result to any Coxeter group.
Related
- Hopf bimodules for bialgebroids
- Classification of Finite-Dimensional Simple Novikov Algebras and Their Irreducible Bimodules in Positive Characteristic
- Simple Harish-Chandra bimodules in positive characteristics
- Graded bimodules over split group algebras: stabilizers and Peirce decompositions
- Banach Gelfand Triples with Kernel Theorems via Equivalence Bimodules
BibTeX
@misc{xie2026parabolic,
title = {{Parabolic filtrations of Soergel bimodules}},
author = {Xun Xie and Qifan Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33490},
eprint = {2610.33490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33490},
}
RIS
TY - UNPB
TI - Parabolic filtrations of Soergel bimodules
AU - Xun Xie
AU - Qifan Zhou
PY - 2026
DA - 2026/10/24
PB - arXiv
DO - 10.48550/arxiv.2610.33490
UR - https://arxiv.org/abs/2610.33490
AB - Given a Coxeter system (W,S) and a subset I⊂ S. Let W_I=⟨ I⟩ be the standard parabolic subgroup of a Coxeter group W, and ^IW be the set of representatives of minimal length in the coset W_Ibackslash W. We construct a parabolic version of Soergel's standard filtration, and show that, given any Soergel bimodule, each subquotient of its parabolic filtration is a direct sum of grading shifts of B_x^I⊗_R R_y with x∈ W_I and y∈ ^IW. By applying the character theorem of Elias and Williamson, we deduce the positivity of coefficients in the expansion of C_w with respect to the hybrid basis {C_xT_y:x∈ W_I,y∈ ^IW}. This generalizes Grojnowski and Haiman's result to any Coxeter group.
ER -