Filtrations on block subalgebras of reduced enveloping algebras

We study the interaction between the block decompositions of reduced enveloping algebras in positive characteristic, the Poincaré-Birkhoff-Witt (PBW) filtration, and the nilpotent cone. We provide two natural versions of the PBW filtration on the block subalgebra [Formula: see text] of the restricted universal enveloping algebra [Formula: see text] and show these are dual to each other. We also consider a shifted PBW filtration for which we relate the associated graded algebra to the algebra of functions on the Frobenius neighborhood of [Formula: see text] in the nilpotent cone and the coinvariants algebra corresponding to [Formula: see text]. In the case of [Formula: see text] in characteristic [Formula: see text] we determine the associated graded algebras of these filtrations on block subalgebras of [Formula: see text]. We also apply this to determine the structure of the adjoint representation of [Formula: see text].

Irreducible Components of the Global Nilpotent Cone

This paper gives a combinatorial description of the set of irreducible components of the semistable locus of the global nilpotent cone, in genus $\ge 2$.

The nilpotent cone in the Mukai system of rank two and genus two

We study the nilpotent cone in the Mukai system of rank two and genus two. We compute the degrees and multiplicities of its irreducible components and describe their cohomology classes.

Calculations with graded perverse-coherent sheaves

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.