Calculations with graded perverse-coherent sheaves

TL;DR

This paper computes properties of graded perverse-coherent sheaves on the nilpotent cone, confirming predictions and explicitly describing simple objects for PGL_3, with applications to quantum group cohomology.

Contribution

It provides explicit calculations of weights and simple objects in graded perverse-coherent sheaves, confirming Ostrik's prediction and describing all simples for PGL_3 in most characteristics.

Findings

Confirmed Ostrik's prediction on weights.

Explicitly described all simple sheaves for PGL_3.

Proved non-existence of positive grading in certain cases.

Abstract

In this paper, we carry out several computations involving graded (or $\mathbb{G}_{\mathrm{m}}$-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 $\mathbb{G}_{\mathrm{m}}$-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 $\mathsf{PCoh}^{\mathbb{G}_{\mathrm{m}}}(\mathcal{N})$ never admits a positive grading when the characteristic of the field is greater than 3.