The dictionary for characters of smooth group schemes

We describe a function-sheaf dictionary for one-dimensional characters of arbitrary smooth group schemes over finite fields. In previous work, we considered the case of commutative smooth group schemes and found that the standard definition of character sheaves produced a dictionary with a nontrivial kernel. In this paper we give a modification of the category of character sheaves that remedies this defect, and is also extensible to non-commutative groups. As an application, we use these emph{commutative character sheaves} to geometrize types for supercuspidal representations of $p$-adic groups.