Kapranov and Vasserot, and then, later, Bridgeland, King and Reid showed this correspondence can be recast and extended as an equivalence of derived categories of coherent sheaves. When this framework is extended to finite subgroups of $\mathrm{GL}(2,\mathbb{C})$ generated by reflections, the equivalence of categories becomes a semiorthogonal decomposition whose components are, conjecturally, in bijection with irreducible representations of $G$. This correspondence has been verified in recent work of Potter and of Capellan for a particular embedding of the dihedral groups $D_n$ in $\mathrm{GL}(2,\mathbb{C})$. I will discuss recent joint work verifying this decomposition in further cases.

I will present some forthcoming work with Thomas Brazelton, in which we compute $\mathbb{A}^1$-degrees of Wronski maps with twisted real structures. Wronski maps arise in Schubert calculus and moduli of curves and have rich enumerative properties over both R and C; they measure ramification points of linear series on $\mathbb{P}^1$. By varying whether the ramification points are real or complex conjugate pairs, we vary the real structure of a Wronski-type family over $\overline{M_{0,n}}$. We describe how its $\mathbb{A}^1$-degree changes as the real structure is twisted.