Faces of polytopes and Koszul algebras

Let $\g$ be a reductive Lie algebra and $V$ a $\g$-semisimple module. In this article, we study the category $\G$ of graded finite-dimensional representations of $\g \ltimes V$. We produce a large class of truncated subcategories, which are directed and highest weight. Suppose $V$ is finite-dimensional with weights $\wt(V)$. Let $\Psi \subset \wt(V)$ be the set of weights contained in a face $\F$ of the polytope that is the convex hull of $\wt(V)$. For each such $\Psi$, we produce quasi-hereditary Koszul algebras. We use these Koszul algebras to construct an infinite-dimensional graded subalgebra $\spg$ of the locally finite part of the algebra of invariants $(END{\C} (\V) \otimes \Sym V)^{\g}$, where $\V$ is the direct sum of all simple finite-dimensional $\g$-modules. We prove that $\spg$ is Koszul of finite global dimension.