Imaginary Highest-weight Representation Theory and Symmetric Functions

Affine Lie algebras admit non-classical highest-weight theories through alternative partitions of the root system. Although significant inroads have been made, much of the classical machinery is inapplicable in this broader context, and some fundamental questions remain unanswered. In particular, the structure of the reducible objects in non-classical theories has not yet been fully understood. This question is addressed here for affine sl(2), which has a unique non-classical highest-weight theory, termed “imaginary”. The reducible Verma modules in the imaginary theory possess an infinite descending series, with all factors isomorphic to a certain canonically associated module, the structure of which depends upon the highest weight. If the highest weight is non-zero, then this factor module is irreducible, and conversely. This paper examines the degeneracy of the factor module of highest-weight zero. The intricate structure of this module is understood via a realization in terms of the symmetric functions. The realization permits the description of a family of singular (critical) vectors, and the classification of the irreducible subquotients. The irreducible subquotients are characterized as those modules with an action given in terms of exponential functions, in the sense of Billig and Zhao.