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), p. 845-873 / Harvested from Numdam

Résumé

Nous calculons l’unique groupe de cohomologie ne s’annulant pas d’un ${\cal O}_{G^0/P}$-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 $G\ne {\bf P}(m),S{\bf P}(m)$ 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.

Abstract

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

Détail

Publié le : 1989-01-01

MR 91k:14036 | Zbl 0667.14023

DOI : https://doi.org/10.5802/aif.1192

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