he antispherical functor $\operatorname{ASph}$ decomposes as a product of functors $\operatorname{ASph}^{\gamma}$ for $\gamma\in \bbT^{\vee}/W_0$, and accordingly the previous diagram decomposes over $\bbT^{\vee}/W_0$. In particular for $\gamma=\{0\}$ we have the commutative diagram $$\xymatrix{ &&\operatorname{Mod}(\cH^{\{0\}}_{\overline{\bbF}_q})\ar[d]^{P^{\{0\}}}  \ar[dr]^{\operatorname{Res}^{\cH^{\{0\}}_{\overline{\bbF}_q}}_{\cH^{\operatorname{sph}}_{\overline{\bbF}_q}}} \