Abstract
This is a survey of some recent developments in the highest weight repesentation theory of the general linear Lie superalgebra \(\mathfrak{g}\mathfrak{l}_{n\vert m}(\mathbb{C})\). The main focus is on the analog of the Kazhdan–Lusztig conjecture as formulated by the author in 2002, which was finally proved in 2011 by Cheng, Lam and Wang. Recently another proof has been obtained by the author joint with Losev and Webster, by a method which leads moreover to the construction of a Koszul-graded lift of category \(\mathcal{O}\) for this Lie superalgebra.
Research supported in part by NSF grant no. DMS-1161094.
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Notes
- 1.
We follow the convention of adding a dot to all q-analogs to distinguish them from their classical counterparts.
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Acknowledgements
Special thanks go to Catharina Stroppel for several discussions which influenced this exposition. I also thank Geoff Mason, Ivan Penkov and Joe Wolf for providing me the opportunity to write a survey article of this nature. In fact I gave a talk on exactly this topic at the Seminar “Lie Groups, Lie Algebras and their Representations” at Riverside in November 2002, when the super Kazhdan–Lusztig conjecture was newborn.
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Brundan, J. (2014). Representations of the General Linear Lie Superalgebra in the BGG Category \(\mathcal{O}\) . In: Mason, G., Penkov, I., Wolf, J. (eds) Developments and Retrospectives in Lie Theory. Developments in Mathematics, vol 38. Springer, Cham. https://doi.org/10.1007/978-3-319-09804-3_3
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