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.

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.

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.

@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},
     pages = {845--873},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {39},
     number = {4},
     year = {1989},
     doi = {10.5802/aif.1192},
     zbl = {0667.14023},
     mrnumber = {91k:14036},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1192/}
}
TY  - JOUR
AU  - Penkov, Ivan
AU  - Serganova, Vera
TI  - Cohomology of $G/P$ for classical complex Lie supergroups $G$ and characters of some atypical $G$-modules
JO  - Annales de l'Institut Fourier
PY  - 1989
SP  - 845
EP  - 873
VL  - 39
IS  - 4
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1192/
DO  - 10.5802/aif.1192
LA  - en
ID  - AIF_1989__39_4_845_0
ER  - 
%0 Journal Article
%A Penkov, Ivan
%A Serganova, Vera
%T Cohomology of $G/P$ for classical complex Lie supergroups $G$ and characters of some atypical $G$-modules
%J Annales de l'Institut Fourier
%D 1989
%P 845-873
%V 39
%N 4
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1192/
%R 10.5802/aif.1192
%G en
%F 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, Tome 39 (1989) no. 4, pp. 845-873. doi : 10.5802/aif.1192. https://aif.centre-mersenne.org/articles/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. | MR | Zbl

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

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

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

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

[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). | MR | Zbl

[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). | MR | Zbl

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

[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. | MR | Zbl

[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). | EuDML | MR | Zbl

[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.

Cité par Sources :