Sur les représentations de dimension finie de la super algèbre de Lie 𝔤𝔩(m,n)  [ On finite dimensional representations of the Lie superalgebra 𝔤𝔩(m,n) ]
Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque no. 311  (2007), Talk no. 963, p. 321-340

The category of the finite dimensional representations of the Lie superalgebra 𝔤𝔩(m,n) is not semi-simple. It has a decomposition in infinitely many blocks and we have been wanting to understand their structure since V. Kac’s work, in 1977. Vera Serganova gives an answer to this problem which is almost complete. She makes use of the ideas developped by Bernstein-Bernstein-Gelfand when they studied the category 𝒪 for the classical case. Since there are no analogues of Kostant and Borel-Weil-Bott theorems in the supercase, she works with the so-called “geometric induction” in order to prove her theorems.

La catégorie des modules de dimension finie sur la super algèbre de Lie 𝔤𝔩(m,n) n’est pas semi-simple. Elle se décompose en une infinité de blocs, dont on cherche depuis les travaux de Kac en 1977 à comprendre la structure. Vera Serganova apporte une réponse presque complète à ce problème, formulée selon le cercle d’idées introduites par Bernstein, Gelfand et Gelfand pour étudier la catégorie 𝒪 dans le cas classique ; ne disposant pas pour 𝔤𝔩(m,n) d’analogues des théorèmes de Kostant et de Borel-Weil-Bott, elle utilise pour démontrer ses résultats des méthodes d’“induction géométrique”.

Classification:  17B10,  14M15,  20G05
Keywords: representations of Lie superalgebras, character formula
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     author = {Gruson, Caroline},
     title = {Sur les repr\'esentations de dimension finie de la super alg\`ebre de Lie $\mathfrak {gl}(m,n)$},
     booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {311},
     year = {2007},
     note = {talk:963},
     pages = {321-340},
     zbl = {1200.17005},
     mrnumber = {2359048},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2005-2006__48__321_0}
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Gruson, Caroline. Sur les représentations de dimension finie de la super algèbre de Lie $\mathfrak {gl}(m,n)$, in Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Talk no. 963, pp. 321-340. http://www.numdam.org/item/SB_2005-2006__48__321_0/

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