Bases of Antidominant Highest Weight Representations

Following the approach of Feigin, Fourier and Littlemann, we explicitly construct a certain family of combinatorial bases (called Vinberg bases) of a special class of highest weight representations of the infinite general linear Lie algebra. These representations, which we call anti-dominant, are characterized by the fact that their highest weights are finite sums of fundamental weights with non-positive coefficients, which can be thought of as the ''reflected'' version of the usual dominant case. The statements and proofs involve special ''skew'' modifications of the technical tools used by the cited authors, which suggests that the natures of dominant and anti-dominant representations are opposite in some sense. In addition, we describe the set of relations of the associated graded module of an anti-dominant representation with respect to the PBW filtration.