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

Ivan Penkov; Vera Serganova

Annales de l'institut Fourier (1989)

  • Volume: 39, Issue: 4, page 845-873
  • ISSN: 0373-0956

Abstract

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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 ) , S 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.

How to cite

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Penkov, Ivan, and Serganova, Vera. "Cohomology of $G/P$ for classical complex Lie supergroups $G$ and characters of some atypical $G$-modules." Annales de l'institut Fourier 39.4 (1989): 845-873. <http://eudml.org/doc/74859>.

@article{Penkov1989,
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.},
author = {Penkov, Ivan, Serganova, Vera},
journal = {Annales de l'institut Fourier},
keywords = {cohomology group of quotient of classical Lie supergroup},
language = {eng},
number = {4},
pages = {845-873},
publisher = {Association des Annales de l'Institut Fourier},
title = {Cohomology of $G/P$ for classical complex Lie supergroups $G$ and characters of some atypical $G$-modules},
url = {http://eudml.org/doc/74859},
volume = {39},
year = {1989},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 39
IS - 4
SP - 845
EP - 873
AB - 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.
LA - eng
KW - cohomology group of quotient of classical Lie supergroup
UR - http://eudml.org/doc/74859
ER -

References

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  13. [13] I. B. PENKOV, Geometric representation theory of complex classical Lie supergroups, Asterisque, to appear. 
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  15. [14] A. N. SERGEEV, The centre of enveloping algebra for Lie superalgebra Q (n, ℂ), Lett. Math. Phys., 7 (1983), 177-179. Zbl0539.17003MR85i:17004
  16. [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). Zbl0573.17002MR85h:17010
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  18. [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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