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Macdonald polynomials and BGG reciprocity for current algebras. (English) Zbl 1367.17017

Summary: 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 \(\mathfrak{sl}_{n+1}\).

MSC:

17B65 Infinite-dimensional Lie (super)algebras
81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
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