, the 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}(\tilde{\cH}^{