Tilting Modules in Truncated Categories

We begin the study of a tilting theory in certain truncated categories of modules $\mathcal G(Γ)$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $Γ= P^+ \times J$, $J$ is an interval in $\mathbb Z$, and $P^+$ is the set of dominant integral weights of the simple Lie algebra. We use this to put a tilting theory on the category $\mathcal G(Γ')$ where $Γ' = P' \times J$, where $P'\subseteq P^+$ is saturated. Under certain natural conditions on $Γ'$, we note that $\mathcal G(Γ')$ admits full tilting modules.

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