It is well known that the Bernstein-Gelfand-Gelfand category O for a Lie algebra g behaves roughly like the Fukaya category of the cotangent bundle to the flag variety G/B. In "Morse theory and tilting sheaves" Nadler defined a geometric construction of tilting modules in category O by taking certain constructible sheaves known as "Morse kernels" on the flag variety and flowing them along a C^*-action whose ascending manifold stratification coincides with the Schubert stratification.
Braden-Licata-Proudfoot-Webster have defined a version of category O for any suitable holomorphic symplectic variety. Other than cotangent bundles the simplest family of holomorphic symplectic varieties are the hypertoric varieties of Bielawski-Dancer. I will describe a construction that generalizes Nadler's result to hypertoric category O. In this case, the "Morse kernels" are certain GKZ hypergeometric systems. This construction arose from joint work with Bullimore, Dimofte, and Gaiotto on mirror symmetry of 3d N=4 gauge theories.