Borel-Weil Theorem for Algebraic Supergroups

We study the structure of an algebraic supergroup $\mathbb{G}$ and establish the Borel-Weil theorem for $\mathbb{G}$ to give a systematic construction of all simple supermodules over an arbitrary field. Especially when $\mathbb{G}$ has a distinguished parabolic super-subgroup, we show that the set of all simple supermodules of $\mathbb{G}$ is parameterized by the set of all dominant weights for the even part of $\mathbb{G}$, prove a super-analogue of the Kempf vanishing theorem, and give a description of Euler characteristics.