Tilting modules for the current algebra of a simple Lie algebra

The category of level zero representations of current and affine Lie algebras shares many of the properties of other well-known categories which appear in Lie theory and in algebraic groups in characteristic p and in this paper we explore further similarities. The role of the standard and co-standard module is played by the finite-dimensional local Weyl module and the dual of the infinite-dimensional global Weyl module respectively. We define the canonical filtration of a graded module for the current algebra. In the case when $\mathfrak g$ is of type $\mathfrak{sl}_{n+1}$ we show that the well-known necessary and sufficient homological condition for a canonical filtration to be a good (or a $\nabla$-filtration) also holds in our situation. Finally, we construct the indecomposable tilting modules in our category and show that any tilting module is isomorphic to a direct sum of indecomposable tilting modules.