he antispherical map induces a bijection $$\xymatrix{ \operatorname{ASph}^{\gamma}:\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}(k)\ar[r]^>>>>>{\sim} & \{\textrm{simple finite dimensional  left $\cH^{\gamma}_{\overline{\bbF}_q}$-modules}\}/\sim. 