Borel–Weil–Bott theory for classical Lie supergroups

The paper is devoted to a systematic construction of the elements of Borel–Weil–Bott theory in the supercase. The main result is a presentation of the cohomology of typical irreducible $G^0$-sheaves on $G^0/B$, where $G^0$ is the connected component of the identity in a classical complex Lie supergroup and $B\hookrightarrow G^0$ an arbitrary Borel subsupergroup. Also presented are some simple known results concerning the cohomology of irreducible $G^0$-sheaves on $G^0/P$ for a parabolic subsupergroup $P$.