On the superdimension of an irreducible representation of a basic classical lie superalgebra

Abstract

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. © 2011 Springer-Verlag Berlin Heidelberg.