no. 4
Cohomology of G/P for classical complex Lie supergroups G and characters of some atypical G-modules
Annales de l'Institut Fourier, Tome 39 (1989) no. 4, pp. 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 P↪G 0 is an arbitrary parabolic subsupergroup. In particular we prove that for G≠ℙ(m),Sℙ(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 P↪G 0 un sous-supergroupe parabolique arbitraire. En particulier, nous démontrons que pour G≠P(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.

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     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},
     pages = {845--873},
     year = {1989},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {4},
     doi = {10.5802/aif.1192},
     mrnumber = {91k:14036},
     zbl = {0667.14023},
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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, Tome 39 (1989) no. 4, pp. 845-873. doi: 10.5802/aif.1192

[1] J. N. Bernstein, D. A. Leites, A character formula for irreducible finite dimensional modules over the Lie superalgebras of series gl and sl, C. R. Acad. Sci. Bulg., 33 (1980), 1049-1051 (in Russian).

[2] M. Demazure, Une démonstration algébrique d'un théorème de Bott, Inv. Math., 5 (1968), 349-356. | Zbl | MR

[3] V. G. Kac, Lie superalgebras, Adv. Math., 26 (1977), 8-96. | Zbl | MR

[4] V. G. Kac, Characters of typical representations of classical Lie superalgebras, Comm. Alg., 5 (8) (1977), 889-897. | Zbl | MR

[5] V. G. Kac, Representations of classical Lie superalgebras, Lect. Notes in Math., 676 (1978), 597-626. | Zbl | MR

[6] V. G. Kac, Laplace operators of infinite-dimensional Lie algebras and theta functions, Proc. Nat. Acad. Sci. USA, 81 (1984), 645-647. | Zbl | MR

[7] Yu I. Manin, Gauge fields and complex geometry, Nauka, Moscow, 1984 (in Russian).

[8] D. A. Leites, Character formulas for irreducible finite dimensional modules of simple Lie superalgebras, Funct. Anal. i ego Pril. 14 (1980), N° 2, 35-38 (in Russian). | Zbl | MR

[9] D. A. Leites, Lie superalgebras, Sovr. Probl. Mat. 25, VINITI, Moscow, 1984, 3-47 (in Russian). | Zbl

[10] D. A. Leites (Ed), Seminar on supermanifolds N° 8, Stockholm University preprint, 1986.

[11] I. B. Penkov, Characters of typical irreducible finite dimensional q(m)-modules, Funct. Anal. i Pril., 20, N° 1 (1986), 37-45 (in Russian). | Zbl | MR

[12] I. B. Penkov, Borel-Weil-Bott theory for classical Lie supergroups, Sovr. Probl. Mat. 32, VINITI, Moscow, (1988), 71-124 (in Russian). | Zbl | MR

[13] I. B. Penkov, Geometric representation theory of complex classical Lie supergroups, Asterisque, to appear.

[13'] I. B. Penkov, Classical Lie supergroups and Lie superalgebras and their representations, Preprint of Institut Fourier, 1988 (contains a preliminary version of Chapters 0, 1, 2 of [13]).

[14] A. N. Sergeev, The centre of enveloping algebra for Lie superalgebra Q (n, ℂ), Lett. Math. Phys., 7 (1983), 177-179. | Zbl | MR

[15] A. N. Sergeev, The tensor algebra of the standard representation as a module over the Lie superalgebra gl(m/n) and Q (n), Mat. Sbornik, 123 (165) (1984), N° 3, 422-430 (in Russian). | Zbl | MR | EuDML

[16] J. Thierry-Mieg, Irreducible representations of the basic classical Lie superalgebras SU(m/n)/U(1), OSP(m/2n), D(2/1,∞), G(3), F(4), Group Theoretical Methods in Physics, Lect. Notes in Phys., 201 (1984), 94-98.

[17] J. Thierry-Mieg, Tables of irreducible representations of the basic classical Lie superalgebras, Preprint of Groupe d'astrophysique relativiste CNRS, Observatoire de Meudon, 1985.

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