Typical representations of admissible type I algebraic supergroups

We study typical representations of an admissible type I algebraic supergroup $G$. We prove that, for a dominant weight $\lambda$, the induced supermodule $H_G^0(\lambda )$ is simple if and only if $\lambda$ is typical and the induced module $H_{G_{\mathrm{e}\mathrm{v}}}^0(\lambda )$ for the even subgroup $G_{\mathrm{e}\mathrm{v}}$ is simple. We establish equivalences between typical blocks for $G$ and blocks for $G_{\mathrm{e}\mathrm{v}}$, and give a projectivity criterion for Kac modules over every Frobenius kernel. For connected $SpO(2\mathrm{\ell }\mathrm{\mid }2)$ over an algebraically closed field of odd characteristic, we classify all simple projective supermodules over Frobenius kernels.