Commutative Character Sheaves

In the 1980s, Lusztig defined character sheaves for connected reductive groups over finite fields as a way to study their representation theory.  This theory begins with the sheaf-function dictionary for commutative connected algebraic groups.  I will describe this dictionary, and then generalize to non-connected smooth group schemes, where we need to impose an additional condition on the sheaves in order to recover an isomorphism.  This generalization is useful for studying characters of p-adic tori, and is joint work with Clifton Cunningham.