 The singular locus of the parametrizing $k$-scheme $$\big((\bbT^{\vee}\times V_{\mathbf{\whT},0})/W_0\big)_{\gamma}$$ is given by $(0,0)\times\bbG_m\subset V_{\mathbf{\whT},0}$ in the standard coordinates, and its $k$-points correspond to the supersingular Hecke modules through the correspondence $\operatorname{ASph}^{\gamma}$.  