Demazure Modules, Fusion Products and Q-Systems

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In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the current algebra associated to a simple Lie algebra. These modules are indexed by an \({|R^{+}|}\)-tuple of partitions \({{\mathbf \xi}=(\xi^\alpha)}\), where α varies over a set \({R^{+}}\) of positive roots of \({\mathfrak{g}}\) and we assume that they satisfy a natural compatibility condition. In the case when the \({\xi^\alpha}\) are all rectangular, for instance, we prove that these modules are Demazure modules in various levels. As a consequence, we see that the defining relations of Demazure modules can be greatly simplified. We use this simplified presentation to relate our results to the fusion products, defined in (Feigin and Loktev in Am Math Soc Transl Ser (2) 194:61–79, 1999), of representations of the current algebra. We prove that the Q-system of (Hatayama et al. in Contemporary Mathematics, vol. 248, pp. 243–291. American Mathematical Society, Providence, 1998) extends to a canonical short exact sequence of fusion products of representations associated to certain special partitions \({\xi}\).Finally, in the last section we deal with the case of \({\mathfrak{sl}_2}\)and prove that the modules we define are just fusion products of irreducible representations of the associated current algebra and give monomial bases for these modules.