  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.
