Macdonald polynomials and BGG reciprocity for current algebras

Abstract

We study the category \(\mathcal I _{\mathrm{gr }}\) of graded representations with finite-dimensional graded pieces for the current algebra \(\mathfrak{g }\otimes \mathbf{C }[t]\) where \(\mathfrak{g }\) is a simple Lie algebra. This category has many similarities with the category \(\mathcal O \) of modules for \(\mathfrak{g }\), and in this paper, we prove an analog of the famous BGG duality in the case of \(\mathfrak{sl }_{n+1}\).