Equivariant coherent sheaves on the exotic nilpotent cone

Abstract:

Let $G=Sp_{2n}(\mathbb {C})$, and $\mathfrak {N}$ be Kato’s exotic nilpotent cone. Following techniques used by Bezrukavnikov in 2003 to establish a bijection between $\boldsymbol {\Lambda }^+$, the dominant weights for an arbitrary 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 us to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category $D^b(\mathrm {Coh}^G(\mathfrak {N}))$, and deduce that the resulting $t$-structure on $D^b(\mathrm {Coh}^G(\mathfrak {N}))$ 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 the $t$-structure on $D^b(\mathrm {Coh}^G(\mathfrak {N}))$.