Parabolic filtrations of Soergel bimodules

Xun Xie; Qifan Zhou

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arXiv · 2610.33490

Parabolic filtrations of Soergel bimodules

Xun Xie·Qifan Zhou

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Abstract

Given a Coxeter system $(W,S)$ and a subset $I\subset S$. Let $W_I=\langle I\rangle$ be the standard parabolic subgroup of a Coxeter group $W$, and ${}^IW$ be the set of representatives of minimal length in the coset $W_I\backslash W$. 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 $B_x^I\otimes_R R_y$ with $x\in W_I$ and $y\in {}^IW$. 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 $\{C_xT_y:x\in W_I,y\in {}^IW\}$. This generalizes Grojnowski and Haiman's result to any Coxeter group.

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BibTeX RIS

Xun Xie, Qifan Zhou. 2026-10-24. Parabolic filtrations of Soergel bimodules. https://arxiv.org/abs/2610.33490

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