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

Penkov, Ivan; Serganova, Vera

Annales de l'Institut Fourier, Tome 39 (1989) no. 4, pp. 845-873.

Nous calculons l’unique groupe de cohomologie ne s’annulant pas d’un ${𝒪}_{G^0/P}$-module $G^0$-linéarisé localement libre générique, où $G^0$ est la composante d’identité d’un supergroupe de Lie $G$ classique complexe et $P\hookrightarrow G^0$ un sous-supergroupe parabolique arbitraire. En particulier, nous démontrons que pour $G\ne \mathbf{P}\left(m\right),S\mathbf{P}\left(m\right)$ ce groupe de cohomologie est un $G^0$-module irréductible. Comme application, nous généralisons la formule de caractère des $G^0$-modules irréductibles typiques à une classe naturelle des modules atypiques apparaissant de cette manière.

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.

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