\lra\overline{\bbF}_q,$$ and then $\operatorname{ASph}^{\gamma}(v)$ identifies with the central reduction $$\cA_{0,\theta_v}:=\cA_{0,\overline{\bbF}_q}\otimes_{Z(\cH_{0,\overline{\bbF}_q}),\theta_v}\overline{\bbF}_q$$ of the mod $p$ regular antispherical representation $\sA_{0,\overline{\bbF}_q}$ specializing \[sA0q\]. 