  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.
