Japanese Journal of Mathematics

, Volume 10, Issue 2, pp 135–235

Characters of (relatively) integrable modules over affine Lie superalgebras

Original Article

DOI: 10.1007/s11537-015-1464-2

Cite this article as:
Gorelik, M. & Kac, V.G. Jpn. J. Math. (2015) 10: 135. doi:10.1007/s11537-015-1464-2


In the paper we consider the problem of computation of characters of relatively integrable irreducible highest weight modules L over finite-dimensional basic Lie superalgebras and over affine Lie superalgebras \({\mathfrak{g}}\). The problem consists of two parts. First, it is the reduction of the problem to the \({\overline{\mathfrak{g}}}\)-module F(L), where \({\overline{\mathfrak{g}}}\) is the associated to L integral Lie superalgebra and F(L) is an integrable irreducible highest weight \({\overline{\mathfrak{g}}}\)-module. Second, it is the computation of characters of integrable highest weight modules. There is a general conjecture concerning the first part, which we check in many cases. As for the second part, we prove in many cases the KW-character formula, provided that the KW-condition holds, including almost all finite-dimensional \({\mathfrak{g}}\)-modules when \({\mathfrak{g}}\) is basic, and all maximally atypical non-critical integrable \({\mathfrak{g}}\)-modules when \({\mathfrak{g}}\) is affine with non-zero dual Coxeter number.

Keywords and phrases

basic Lie superalgebra defect dual Coxeter number affine Lie superalgebra odd reflection integrable highest weight module over basic and affine Lie superalgebra maximally atypical module KW-condition Enright functor relatively integrable module character formula 

Mathematics Subject Classification (2010)

17B10 05E10 

Copyright information

© The Mathematical Society of Japan and Springer Japan 2015

Authors and Affiliations

  1. 1.Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael
  2. 2.Department of Mathematics, 2-178Massachusetts Institute of TechnologyCambridgeUSA