  We study the representation theory of the invariant subalgebra of the Weyl
algebra under a torus action, which we call a "hypertoric enveloping algebra."
We define an analogue of BGG category O for this algebra, and identify it with
a certain category of sheaves on a hypertoric variety. We prove that a regular
block of this category is highest weight and Koszul, identify its Koszul dual,
compute its center, and study its cell structure. We also consider a collection
of derived auto-equivalences analogous to the shuffling and twisting functors
for BGG category O.
