Reciprocity in character theory of finite semigroups

This paper concerns the complex characters of finite semigroups. We begin by reducing the general problem to the study of local monoids M=Gunion or logical sumJunion or logical sum{0}, where G is the unit group and J is a regular image-class. Let H be a maximal subgroup of J. For a image-module U with character θ, let image be the associated image-module and let image be the character of the image-module image. For a image-module V with character ψ, we construct a natural image-module image with character image. We prove the reciprocity theorem: image. As a consequence we find a characterization of the J-cuspidal characters of G. We go on to interpret our results for groups, introducing the concept of double parabolic induction.