Koszul duality between Higgs and Coulomb categories $\mathcal{O}$

Understand

We prove a Koszul duality theorem between the category of weight modules over the quantized Coulomb branch (as defined by Braverman, Finkelberg and Nakajima) attached to a group $G$ and representation $V$ and a category of $G$-equivariant D-modules on the vector space $V$.

• This is proven by relating both categories to an explicit, combinatorially presented category.
• These categories are related to generalized categories $\mathcal{O}$ for symplectic singularities.
• Letting $\mathcal{O}_{\operatorname{Coulomb}}$ and $\mathcal{O}_{\operatorname{Higgs}}$ be these categories for the Coulomb and Higgs branches associated to $V$ and $G$, we obtain a functor $\mathcal{O}_{\operatorname{Coulomb}}^!\to \mathcal{O}_{\operatorname{Higgs}}$ from the Koszul dual of one to the other.