On the Superdimension of an Irreducible Representation of a Basic Classical Lie Superalgebra
In this paper we prove the Kac-Wakimoto conjecture that a simple module over a basic classical Lie superalgebra has non-zero superdimension if and only if it has maximal degree of atypicality. The proof is based on the results of [Duflo and Serganova, On associated variety for Lie superalgebras, math/0507198] and [Gruson and Serganova, Proceedings of the London Mathematical Society, doi:10.1112/plms/pdq014].We also prove the conjecture in [Duflo and Serganova, On associated variety for Lie superalgebras, math/0507198] about the associated variety of a simple module and the generalized Kac-Wakimoto conjecture in [Geer, Kujawa and Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories, arXiv:1001.0985v1] for the general linear Lie superalgebra.
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The author thanks Michel Duflo and Jonathan Kujawa for stimulating discussions and the referee for helpful suggestions. This work was partially supported by NSF grant 0901554.
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