Quasi-split generalization of the spherical-module theorem for rational Cherednik algebras

Let $G$ be a quasi-split group in the generality considered in the paper, let $a\in\mathfrak{c}(F)^{\textup{rs}}_{\nu}$, and use the perverse and Chern filtrations, the affine Springer fiber $\textup{Sp}_{a}$, the group $B_a$, and the rational Cherednik algebras $\mathfrak{H}^{\textup{rat}}_{\nu}$ and $\mathfrak{H}^{\textup{rat}}_{\nu,\epsilon=1}$ as in Theorem

$$. **Quasi-split generalization conjecture.** The statements of Theorem$$

hold for quasi-split groups $G$ (in the generality of $\S$). The theorem is proved for split groups, while the paper proves only a weaker version for quasi-split non-split groups; extending the geometric argument to this setting remains open.