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We compute the unique nonzero cohomology group of a generic $G^0$- linearized locally free $𝒪$-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 $G\ne \mathbb{P}\left(m\right),S\mathbb{P}\left(m\right)$ 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.

Let $𝔤$ be a complex reductive Lie algebra and $𝔨\subset 𝔤$ be any reductive in $𝔤$ subalgebra. We call a $(𝔤,𝔨)$-module $M$ bounded if the $𝔨$-multiplicities of $M$ are uniformly bounded. In this paper we initiate a general study of simple bounded $(𝔤,𝔨)$-modules. We prove a strong necessary condition for a subalgebra $𝔨$ to be bounded (Corollary 4.6), to admit an infinite-dimensional simple bounded $(𝔤,𝔨)$-module, and then establish a sufficient condition for a subalgebra $𝔨$ to be bounded (Theorem 5.1). As a result we are able to classify...