Calculations with graded perverse-coherent sheaves

Abstract

In this paper, we carry out several computations involving graded (or Gm-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 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 = P GL 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 PCoh Gm (N ) never admits a positive grading when the characteristic of the field is greater than 3. P.A. was supported by NSF Grant No. DMS-1500890. Preliminaries on nilpotent orbits2.1. Notation and conventions. Let k be an algebraically closed field. For a graded k-vector space V = m∈Z V m , we define V n to be the graded k-vector space given by (V n ) m = V m+n . It is sometimes convenient to think of V as a

Silting complexes of coherent sheaves and the Humphreys conjecture

“…Moreover, its action on coherent sheaves on an orbit can be described explicitly described using [ Proof. The analogous statement for irreducible representations of G m ˙GxC is shown in the proof of [3,Theorem 4.5] (see also [3,Remark 4.6]). From this, one sees that if V is the costandard G xC red -module with simple socle L, then V idˆσC has simple socle L ˚, and furthermore it has the same composition factors the appropriate costandard module.…”

“…According to [3,Corollary 4.2], this functor preserves supports. Moreover, its action on coherent sheaves on an orbit can be described explicitly described using [ Proof.…”

“…(A priori, the image might have been of the form ICpC, L ω qxny for some n P Z; this integer was determined to be 0 in [3].) The resulting map…”

“…This turns out to be false: the calculations in [3] yield counterexamples when G " PGL 3 , and C is the subregular nilpotent orbit. However, (7.4) is (trivially) true when C is the zero nilpotent orbit.…”

Integral exotic sheaves and the modular Lusztig–Vogan bijection

“…(One exception is the proof of Proposition 5.2, where the $$\mathbb {G}_{\mathrm{m}}$$‐action plays an important role.) However, there is no loss in doing so: as explained in [5, Section 3], the graded Lusztig–Vogan bijection is completely determined by the ordinary (ungraded) Lusztig–Vogan bijection. In particular, the main theorem of this paper implies that the graded Lusztig–Vogan bijection is also independent of $$\mathbb {k}$$.…”