Borel-Weil theorem for algebraic supergroups

Abstract

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