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New graded modules for the current algebra of $\mathfrak{sl}_n$ areintroduced. Relating these modules to the fusion product of simple$\mathfrak{sl}_n$-modules and local Weyl modules of truncated current algebrasshows their expected impact on several outstanding conjectures. We furthergeneralize results on PBW filtrations of simple $\mathfrak{sl}_n$-modules anduse them to provide decomposition formulas for these new modules in importantcases.

New homogeneous ideals for current algebras: filtrations, fusion products and Pieri rules