Equivariant coherent sheaves on the exotic nilpotent cone

TL;DR

This paper establishes a bijection between dominant weights and pairs of nilpotent orbits with irreducible representations in the context of the exotic nilpotent cone, extending techniques from classical theory.

Contribution

It proves a new bijection for the exotic nilpotent cone, showing the category of equivariant coherent sheaves has a perverse coherent t-structure similar to classical cases.

Findings

Vanishing higher cohomology of dominant line bundles on the exotic Springer resolution

Construction of a quasi-exceptional set generating the derived category

Identification of the t-structure with the perverse coherent t-structure

Abstract

Let $G=Sp_{2n}(\mathbb{C})$, and $\mathfrak{N}$ be Kato's exotic nilpotent cone. Following techniques used by Bezrukavnikov in [5] to establish a bijection between $\Lambda^+$, the dominant weights for a simple algebraic group $H$, and $\textbf{O}$, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution $\widetilde{\mathfrak{N}}$ have vanishing higher cohomology, and compute their global sections using techniques of Broer. This allows to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category $D^b(Coh^G(\mathfrak{N}))$, and deduce that the resulting $t$-structure on $D^b(Coh^G(\mathfrak{N}))$ coincides with the perverse coherent $t$-structure. The desired result now follows from the bijection between costandard objects and simple objects in the heart of this $t$-structure on $D^b(Coh^G(\mathfrak{N}))$.