Cohomology of G/P for classical complex Lie supergroups G and characters of some atypical G-modules (Q1117285)

We compute the unique nonzero cohomology group of a generic \(G^ 0\)- linearized locally free \({\mathcal O}_{G^ 0/P}\)-module, where \(G^ 0\) is the identity component of a complex classical Lie supergroup G and \(P\hookrightarrow G^ 0\) is an arbitrary parabolic subsupergroup. In particular we prove that for \(G\neq {\mathbb{P}}(m), S{\mathbb{P}}(m)\) this cohomology group is an irreducible \(G^ 0\)-module. As an application we generalize the character formula of typical irreducible \(G^ 0\)-modules to a natural class of atypical modules arising in this way.