  Motivated by the polynomial representation theory of the general linear group
and the theory of symplectic singularities, we study a category of perverse
sheaves with coefficients in a field $k$ on any affine unimodular hypertoric
variety. Our main result is that this is a highest weight category whose Ringel
dual is the corresponding category for the Gale dual hypertoric variety. On the
way to proving our main result, we confirm a conjecture of Finkelberg-Kubrak in
the case of hypertoric varieties. We also show that our category is equivalent
to representations of a combinatorially-defined algebra, recently introduced in
a related paper.
