Cohomology of G/P for classical complex Lie supergroups G and characters of some atypical G-modules
Annales de l'Institut Fourier, Volume 39 (1989) no. 4, p. 845-873

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 PG 0 is an arbitrary parabolic subsupergroup. In particular we prove that for G(m),S(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.

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 PG 0 un sous-supergroupe parabolique arbitraire. En particulier, nous démontrons que pour GP(m),SP(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.

@article{AIF_1989__39_4_845_0,
     author = {Penkov, Ivan and Serganova, Vera},
     title = {Cohomology of $G/P$ for classical complex Lie supergroups $G$ and characters of some atypical $G$-modules},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {39},
     number = {4},
     year = {1989},
     pages = {845-873},
     doi = {10.5802/aif.1192},
     zbl = {0667.14023},
     mrnumber = {91k:14036},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1989__39_4_845_0}
}
Penkov, Ivan; Serganova, Vera. Cohomology of $G/P$ for classical complex Lie supergroups $G$ and characters of some atypical $G$-modules. Annales de l'Institut Fourier, Volume 39 (1989) no. 4, pp. 845-873. doi : 10.5802/aif.1192. http://www.numdam.org/item/AIF_1989__39_4_845_0/

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