In this paper, we carry out several computations involving graded (or Gm-equivariant) perversecoherent 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 Gm-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 = PGL3, 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 PCohGm(N) never admits a positive grading when the characteristic of the field is greater than 3.
Publication Source (Journal or Book title)
Quarterly Journal of Mathematics
Achar, P., & Hardesty, W. (2019). Calculations with graded perverse-coherent sheaves. Quarterly Journal of Mathematics, 70 (4), 1327-1352. https://doi.org/10.1093/qmath/haz016