Sur les représentations de dimension finie de la super algèbre de Lie \mathrm{𝔤𝔩}(m,n)

Gruson, Caroline

Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 963, pp. 321-340.

La catégorie des modules de dimension finie sur la super algèbre de Lie \mathrm{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 \mathrm{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”.

The category of the finite dimensional representations of the Lie superalgebra \mathrm{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.

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