Character Sheaves on Reductive Lie Algebras in Positive Characteristic

TL;DR

This paper characterizes character sheaves on reductive Lie algebras in large positive characteristic using microlocal methods, establishing criteria based on singular support and quasi-admissibility, with geometric proofs of key equivalences.

Contribution

It provides a microlocal characterization of character sheaves in positive characteristic and offers geometric proofs of their equivalence with admissibility, extending prior work to new settings.

Findings

Character sheaves are identified by nilpotent singular support and quasi-admissibility.

Geometric proofs establish the equivalence between admissibility and being a character sheaf.

Characterizations of cuspidal sheaves are extended following Mirković's work.

Abstract

We prove a microlocal characterisation of character sheaves on a reductive Lie algebra over an algebraically closed field of sufficiently large positive characteristic: a perverse irreducible G-equivariant sheaf is a character sheaf if and only if it has nilpotent singular support and is quasi-admissible. We also present geometric proofs, in positive characteristic, of the equivalence between being admissible and being a character sheaf, and various characterisations of cuspidal sheaves, following the work of Mirkovi\'c.