On Lusztig's conjecture for connected and disconnected exceptional groups

Lusztig conjectured that the almost characters of a finite reductive group are up to a scalar the same as the characteristic functions of the rational character sheaves defined on the corresponding algebraic group. In this paper we verify this conjecture for the uniform almost characters of the Suzuki and the Ree groups, and to show that a conjecture of the same type holds for the disconnected case of type B2. We then state a conjecture on the possible values of the uniform almost characters on unipotent elements in the disconnected case of type F4.