Quantizations of conical symplectic resolutions II: category $\mathcal O$ and symplectic duality

Abstract

We define and study category O for a symplectic resolution, generalizing the classical BGG category O, which is associated with the Springer resolution. This includes the development of intrinsic properties paralleling the BGG case, such as a highest weight structure and analogues of twisting and shuffling functors, along with an extensive discussion of individual examples.We observe that category O is often Koszul, and its Koszul dual is often equivalent to category O for a different symplectic resolution. This leads us to define the notion of a symplectic duality between symplectic resolutions, which is a collection of isomorphisms between representation theoretic and geometric structures, including a Koszul duality between the two categories. This duality has various cohomological consequences, including (conjecturally) an identification of two geometric realizations, due to Nakajima and Ginzburg/Mirković-Vilonen, of weight spaces of simple representations of simply-laced simple algebraic groups.An appendix by Ivan Losev establishes a key step in the proof that O is highest weight.

Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and slices in the affine Grassmannian

“…Nevertheless it is expected that the pushforward from M(λ, µ) and the hyperbolic restriction for χ are exchanged under the duality. It should be also related to the symplectic duality [BLPW16].…”

From hypertoric geometry to bordered Floer homology via the m=1 amplituhedron

“…expectations. One such expectation, which comes from the work of the second author with Braden, Proudfoot and Webster [BLPW16,BLPW10,BLPW12], is a relationship between symplectic duality and Koszul duality. This expectation has been established in the case of hypertoric varieties, as the hypertoric categories O associated to symplectic dual hypertoric varieties are Koszul dual.…”

Note on some properties of generalized affine Grassmannian slices

“…It carries a natural Poisson structure, together with a canonical quantization M q C (µ, λ) with respect to the natural Poisson structure [BFN16b]. It is expected to be symplectic dual to the Higgs branch M H (µ, λ) of the same gauge theory [BPW12,BLPW14], and the symplectic duality has been worked out in many cases when the gauge group is simply-laced [Web16, KTW + 18]. These features make it an ideal test ground for geometric representation theory.…”