Calculations with graded perverse-coherent sheaves

Abstract

Abstract In this paper, we carry out several computations involving graded (or ${{\mathbb {G}}_{\textrm {m}}}$-equivariant) perverse-coherent sheaves on the nilpotent cone of a reductive group in good characteristic. In the first part of the paper, we compute the weight of the ${{\mathbb {G}}_{\textrm {m}}}$-action on certain normalized (or ‘canonical’) simple objects, confirming an old prediction of Ostrik. In the second part of the paper, we explicitly describe all simple perverse-coherent sheaves for $G = PGL_3$, in every characteristic other than 2 or 3. Applications include an explicit description of the cohomology of tilting modules for the corresponding quantum group, as well as a proof that $\textsf {PCoh}^{{{\mathbb {G}}_{\textrm {m}}}}({\mathcal {N}})$ never admits a positive grading when the characteristic of the field is greater than 3.